So far, these are the patristic references to 1 Timothy 4:10 that I have been able to find:
And, indeed, the old Hebrew wanderers in the desert received typically the end of the threatening; for they are said not to have entered into the rest, because of unbelief, till, having followed the successor of Moses, they learned by experience, though late, that they could not be saved otherwise than by believing on Jesus. But the Lord, in His love to man, invites all men to the knowledge of the truth, and for this end sends the Paraclete. What, then, is this knowledge? Godliness; and “godliness,” according to Paul, “is profitable for all things, having the promise of the life that now is, and of that which is to come.”967 If eternal salvation were to be sold, for how much, O men, would you propose to purchase it? Were one to estimate the value of the whole of Pactolus, the fabulous river of gold, he would not have reckoned up a price equivalent to salvation.
Do not, however, faint. You may, if you choose, purchase salvation, though of inestimable value, with your own resources, love and living faith, which will be reckoned a suitable price. This recompense God cheerfully accepts; “for we trust in the living God, who is the Saviour of all men, especially of those who believe.”968 [1 Timothy iv. 10]
Clement of Alexandria, Exhortation to the Heathen, Chapter IX (Source)
And to speak comprehensively, all benefit appertaining to life, in its highest reason, proceeding from the Sovereign God, the Father who is over all, is consummated by the Son, who also on this account “is the Saviour of all men,” says the apostle, “but especially of those who believe.”3494 But in respect of its immediate reason, it is from those next to each, in accordance with the command and injunction of Him who is nearest the First Cause, that is, the Lord.
Clement of Alexandria, Stromata, Book VI, Chapter XVII (Source)
“But,” say they, “God is ‘good,’ and ‘most good,’715 and ‘pitiful-hearted,’ and ‘a pitier,’ and ‘abundant in pitiful-heartedness,’716 which He holds ‘dearer than all sacrifice,’717 ‘not thinking the sinner’s death of so much worth as his repentance’,718 ‘a Saviour of all men, most of all of believers.’719 And so it will be becoming for ‘the sons of God’720 too to be ‘pitiful-hearted’721 and ‘peacemakers;’722 ‘giving in their turn just as Christ withal hath given to us;’723 ‘not judging, that we be not judged.’724 For ‘to his own lord a man standeth or falleth; who art thou, to judge another’s servant?’725 ‘Remit, and remission shall be made to thee.’”726 Such and so great futilities of theirs wherewith they flatter God and pander to themselves, effeminating rather than invigorating discipline, with how cogent and contrary (arguments) are we for our part able to rebut,—(arguments) which set before us warningly the “severity”727 of God, and provoke our own constancy? Because, albeit God is by nature good, still He is “just”728 too.
Tertullian, On Modesty, Chapter II (Source)
But since he has represented those whom he regards as worms, viz., the Christians, as saying that “God, having abandoned the heavenly regions, and despising this great earth, takes up His abode amongst us alone, and to us alone makes His announcements, and ceases not His messages and inquiries as to how we may become His associates for ever,” we have to answer that he attributes to us words which we never uttered, seeing we both read and know that God loves all existing things, and loathes3786 nothing which He has made, for He would not have created anything in hatred. We have, moreover, read the declaration: “And Thou sparest all things, because they are Thine, O lover of souls. For Thine incorruptible Spirit is in all. And therefore those also who have fallen away for a little time Thou rebukest, and admonishest, reminding them of their sins.”3787 How can we assert that “God, leaving the regions of heaven, and the whole world, and despising this great earth, takes up His abode amongst us only,” when we have found that all thoughtful persons must say in their prayers, that “the earth is full of the mercy of the Lord,”3788 and that “the mercy of the Lord is upon all flesh;”3789 and that God, being good, “maketh His sun to arise upon the evil and the good, and sendeth His rain upon the just and the unjust;”3790 and that He encourages us to a similar course of action, in order that we may become His sons, and teaches us to extend the benefits which we enjoy, so far as in our power, to all men? For He Himself is said to be the Saviour of all men, especially of them that believe;3791 and His Christ to be the “propitiation for our sins, and not for ours only, but also for the sins of the whole world.”3792 And this, then, is our answer to the allegations of Celsus. Certain other statements, in keeping with the character of the Jews, might be made by some of that nation, but certainly not by the Christians, who have been taught that “God commendeth His love towards us, in that, while we were yet sinners, Christ died for us;”3793 and although “scarcely for a righteous man will one die, yet peradventure for a good man some would even dare to die.”3794 But now is Jesus declared to have come for the sake of sinners in all parts of the world (that they may forsake their sin, and entrust themselves to God), being called also, agreeably to an ancient custom of these Scriptures, the “Christ of God.”
Origen, Against Celsus, Book IV, Chapter XXVIII (Source)
37. We have lingered over this subject of the martyrs and over the record of those who died on account of pestilence, because this lets us see the excellence of Him who was led as a sheep to the slaughter and was dumb as a lamb before the shearer. For if there is any point in these stories of the Greeks, and if what we have said of the martyrs is well founded — the Apostles, too, were for the same reason the filth of the world and the offscouring of all things, 1 Corinthians 4:13 — what and how great things must be said of the Lamb of God, who was sacrificed for this very reason, that He might take away the sin not of a few but of the whole world, for the sake of which also He suffered? If any one sin, we read, 1 John 2:1-2 We have an advocate with the Father, Jesus Christ the righteous; and He is the propitiation for our sins, and not for ours only, but for those of the whole world, since He is the Saviour of all men, 1 Timothy 4:10 especially of them that believe, who Colossians 2:14-15 blotted out the written bond that was against us by His own blood, and took it out of the way, so that not even a trace, not even of our blotted-out sins, might still be found, and nailed it to His cross; who having put off from Himself the principalities and powers, made a show of them openly, triumphing over them by His cross. And we are taught to rejoice when we suffer afflictions in the world, knowing the ground of our rejoicing to be this, that the world has been conquered and has manifestly been subjected to its conqueror. Hence all the nations, released from their former rulers, serve Him, because He saved the poor from his tyrant by His own passion, and the needy who had no helper. This Saviour, then, having humbled the calumniator by humbling Himself, abides with the visible sun before His illustrious church, tropically called the moon, from generation to generation. And having by His passion destroyed His enemies, He who is strong in battle and a mighty Lord required after His mighty deeds a purification which could only be given Him by His Father alone; and this is why He forbids Mary to touch Him, saying, John 20:17 Touch Me not, for I am not yet ascended to My Father; but go and tell My disciples, I go to My Father and your Father, to My God and your God. And when He comes, loaded with victory and with trophies, with His body which has risen from the dead — for what other meaning can we see in the words, I am not yet ascended to My Father, and I go unto My Father, — then there are certain powers which say, Who is this that comes from Edom, red garments from Bosor; this that is beautiful? Isaiah 63:1 Then those who escort Him say to those that are upon the heavenly gates, Lift up your gates, you rulers, and be lifted up, you everlasting doors, and the king of glory shall come in. But they ask again, seeing as it were His right hand red with blood and His whole person covered with the marks of His valour, Why are Your garments red, and Your clothes like the treading of the full winefat when it is trodden? And to this He answers, I have crushed them. For this cause He had need to wash His robe in wine, and His garment in the blood of the grape. Genesis 49:2 For when He had taken up our infirmities and carried our diseases, and had borne the sin of the whole world, and had conferred blessings on so many, then, perhaps, He received that baptism which is greater than any that could ever be conceived among men, and of which I think He speaks when He says, Luke 12:50 I have a baptism to be baptized with, and how am I straitened till it be accomplished? I enquire here with boldness and I challenge the ideas put forward by most writers. They say that the greatest baptism, beyond which no greater can be conceived, is His passion. But if this be so, why should He say to Mary after it, Touch Me not? He should rather have offered Himself to her touch, when by His passion He had received His perfect baptism. But if it was the case, as we said before, that after all His deeds of valour done against His enemies, He had need to wash His robe in wine, His garment in the blood of the grape, then He was on His way up to the husbandman of the true vine, the Father, so that having washed there and after having gone up on high, He might lead captivity captive and come down bearing manifold gifts — the tongues, as of fire, which were divided to the Apostles, and the holy angels which are to be present with them in each action and to deliver them. For before these economies they were not yet cleansed and angels could not dwell with them, for they too perhaps do not desire to be with those who have not prepared themselves nor been cleansed by Jesus. For it was of Jesus' benignity alone that He ate and drank with publicans and sinners, and suffered the penitent woman who was a sinner to wash His feet with her tears, and went down even to death for the ungodly, counting it not robbery to be equal with God, and emptied Himself, assuming the form of a servant. And in accomplishing all this He fulfils rather the will of the Father who gave Him up for sinners than His own. For the Father is good, but the Saviour is the image of His goodness; and doing good to the world in all things, since God was in Christ reconciling the world to Himself, which formerly for its wickedness was all enemy to Him, He accomplishes His good deeds in order and succession, and does not all at once take all His enemies for His footstool. For the Father says to Him, to the Lord of us all, Sit on My right hand, until I make Your enemies the footstool of Your feet. And this goes on till the last enemy, Death, is overcome by Him. And if we consider what is meant by this subjection to Christ and find an explanation of this mainly from the saying, 1 Corinthians 15:26 When all things shall have been put under Him, then shall the Son Himself be subjected to Him who put all things under Him, then we shall see how the conception agrees with the goodness of the God of all, since it is that of the Lamb of God, taking away the sin of the world. Not all men's sin, however, is taken away by the Lamb of God, not the sin of those who do not grieve and suffer affliction till it be taken away. For thorns are not only fixed but deeply rooted in the hand of every one who is intoxicated by wickedness and has parted with sobriety, as it is said in the Proverbs, Thorns grow in the hand of the drunkard, and what pain they must cause him who has admitted such growth in the substance of his soul, it is hard even to tell. Who has allowed wickedness to establish itself so deeply in his soul as to be a ground full of thorns, he must be cut down by the quick and powerful word of God, which is sharper than a two-edged sword, and which is more caustic than any fire. To such a soul that fire must be sent which finds out thorns, and by its divine virtue stands where they are and does not also burn up the threshing-floors or standing grain. But of the Lamb which takes away the sin of the world and begins to do so by His own death there are several ways, some of which are capable of being clearly understood by most, but others are concealed from most, and are known to those only who are worthy of divine wisdom. Why should we count up all the ways by which we come to believe among men? That is a thing which every one living in the body is able to see for himself. And in the ways in which we believe in these also, sin is taken away; by afflictions and evil spirits and dangerous diseases and grievous sicknesses. And who knows what follows after this? So much as we have said was not unnecessary — we could not neglect the thought which is so clearly connected with that of the words, Behold the Lamb of God that takes away the sin of the world, and had therefore to attend somewhat closely to this part of our subject. This has brought us to see that God convicts some by His wrath and chastens them by His anger, since His love to men is so great that He will not leave any without conviction and chastening; so that we should do what in us lies to be spared such conviction and such chastening by the sorest trials.
38. The reader will do well to consider what was said above and illustrated from various quarters on the question what is meant in Scripture by the word world; and I think it proper to repeat this. I am aware that a certain scholar understands by the world the Church alone, since the Church is the adornment of the world, and is said to be the light of the world. You, he says, Matthew 5:14 are the light of the world. Now, the adornment of the world is the Church, Christ being her adornment, who is the first light of the world. We must consider if Christ is said to be the light of the same world as His disciples. When Christ is the light of the world, perhaps it is meant that He is the light of the Church, but when His disciples are the light of the world, perhaps they are the light of others who call on the Lord, others in addition to the Church, as Paul says on this point in the beginning of his first Epistle to the Corinthians, where he writes, To the Church of God, with all who call on the name of the Lord Jesus Christ. Should any one consider that the Church is called the light of the world, meaning thereby of the rest of the race of men, including unbelievers, this may be true if the assertion is taken prophetically and theologically; but if it is to be taken of the present, we remind him that the light of a thing illuminates that thing, and would ask him to show how the remainder of the race is illuminated by the Church's presence in the world. If those who hold the view in question cannot show this, then let them consider if our interpretation is not a sound one, that the light is the Church, and the world those others who call on the Name. The words which follow the above in Matthew will point out to the careful enquirer the proper interpretation. You, it is said, are the salt of the earth, the rest of mankind being conceived as the earth, and believers are their salt; it is because they believe that the earth is preserved. For the end will come if the salt loses its savour, and ceases to salt and preserve the earth, since it is clear that if iniquity is multiplied and love waxes cold upon the earth, Matthew 24:12 as the Saviour Himself uttered an expression of doubt as to those who would witness His coming, saying, Luke 18:8 When the Son of man comes, shall He find faith upon the earth? then the end of the age will come. Supposing, then, the Church to be called the world, since the Saviour's light shines on it — we have to ask in connection with the text, Behold the Lamb of God, which takes away the sin of the world, whether the world here is to be taken intellectually of the Church, and the taking away of sin is limited to the Church. In that case what are we to make of the saying of the same disciple with regard to the Saviour, as the propitiation for sin? If any man sin, we read, we have an advocate with the Father, Jesus Christ the righteous; and He is the propitiation for our sins, and not for our sins only, but for the sins of the whole world? Paul's dictum appears to me to be to the same effect, when he says, 1 Timothy 4:10 Who is the Saviour of all men, especially of the faithful. Again, Heracleon, dealing with our passage, declares, without any proof or any citation of witnesses to that effect, that the words, Lamb of God, are spoken by John as a prophet, but the words, who takes away the sin of the world, by John as more than a prophet. The former expression he considers to be used of His body, but the latter of Him who was in that body, because the lamb is an imperfect member of the genus sheep; the same being true of the body as compared with the dweller in it. Had he meant to attribute perfection to the body he would have spoken of a ram as about to be sacrificed. After the careful discussions given above, I do not think it necessary to enter into repetitions on this passage, or to controvert Heracleon's careless utterances. One point only may be noted, that as the world was scarcely able to contain Him who had emptied Himself, it required a lamb and not a ram, that its sin might be taken away.
Origen Commentary on John Book VI, 37-38 (Source)
Ver. 10. “For therefore we both labor and suffer reproach, because we trust in the living God, who is the Saviour of all men, specially of them that believe.”
This in effect is to say, wherefore do we mortify ourselves, unless we expect future blessings? Have we endured so many evils, submitted to so many reproaches, suffered such insults and calumnies, and such numerous calamities in vain? For if we did not trust in the living God, on what account did we submit to these things? But if God is here the Saviour1223 of the unbelieving, much more is He of the faithful hereafter. What salvation does he speak of? That to come?1224 “Who is the Saviour,” he says, “of all men, specially of them that believe.” At present he is speaking of that which is here. But how is He the Saviour of the faithful? Had he not been so, they must long since have been destroyed, for all men have made war upon them. He calls him here to endure perils, that having God for his Saviour he may not faint nor need any aid from others, but willingly and with fortitude endure all things. Even those who eagerly grasp at worldly advantages, supported by the hope of gain, cheerfully undertake laborious enterprises.
John Chrysostom, Homilies on 1 Timothy, 1 Timothy 4:1-10, at 1 Timothy 4:10 (Source)
There is also reference in the longer version of Ignatius, Epistle to the Ephesians, Chapter VIII (source) as well as a reference in the spurious Epistle to the Philippians (source) and in the pseudographic "Apostolic Constitutions" from the "Two Ways" section thereof (source)
(This post to be updated if/when needed.)