This set of passages are focused on Paul's (counting Hebrews' as Paul's for simplicity's sake) commands explicit or implicit to give thanks.
The first passage is from Romans 14. This passage mentions that we are to give thanks when we eat. It does so implicitly by assuming that that those who eat give thanks.
The second passage, from 1 Timothy 4, makes the matter more specific. It explains that God has created food to be received with thanks. Furthermore, we are told that the thanksgiving sanctifies the food to us. This provides something of a rationale for the giving of thanks, which we would still do without a reason being provided, but much more so seeing the reason. When you eat, give thanks for the food and thereby render it appropriate for your consumption.
The next passage turns to another topic, namely the salvation of men. We have seen that Paul gives thanks to God for everyone's salvation in a previous segment. In 1 Timothy 2, Paul commands that we not only offer "supplications, prayers, [and] intercessions," but also "giving of thanks," for all men. This reinforces to us the fact that we have a duty to thank God when He answers our prayers of salvation for men. He is the one who changes the heart, and he deserves the thanks when men are saved.
The passage from Colossians 1 has a similar theme. We are to thank God not only for the salvation of others, but for our own salvation as well.
The remaining passages generally commend thanksgiving to us: we are to thank God "always for all things" (Ephesians 5); "in every thing" (Philippians 4; accompanying "whatsoever ye do in word or deed" (Colossians 3); and 1 Thessalonians 5); and "continually" (Hebrews 13). Lastly, two passages in Colossians similarly commend thanksgiving in general without more specific qualification.
Nevertheless, we can still learn additional information about thanksgiving from these passages. In the first of the Colossians passages, we see that thanksgiving abounds out of the root of Christ. Thanksgiving is the fruit of the root of Jesus. We are thankful because of Him.
Finally, the second of the Colossians passages links thanksgiving integrally with prayer. While, of course, there are other ways that we can give thanks (such as by singing psalms), our prayers should not omit thanksgiving. It is an important aspect of prayer to thank God. If possible, we should do so naturally, out of the root of gratitude in us. However, one may train this in oneself by making it a practice never to end one's prayer without thanking God for something, no matter how dire the situation. In this way, one will become accustomed to giving thanks to God unceasingly and it will become more natural.
Him that is weak in the faith receive ye, but not to doubtful disputations. For one believeth that he may eat all things: another, who is weak, eateth herbs. Let not him that eateth despise him that eateth not; and let not him which eateth not judge him that eateth: for God hath received him.
Who art thou that judgest another man's servant? to his own master he standeth or falleth. Yea, he shall be holden up: for God is able to make him stand. One man esteemeth one day above another: another esteemeth every day alike. Let every man be fully persuaded in his own mind.
He that regardeth the day, regardeth it unto the Lord; and he that regardeth not the day, to the Lord he doth not regard it. He that eateth, eateth to the Lord, for he giveth God thanks; and he that eateth not, to the Lord he eateth not, and giveth God thanks.
For none of us liveth to himself, and no man dieth to himself. For whether we live, we live unto the Lord; and whether we die, we die unto the Lord: whether we live therefore, or die, we are the Lord's. For to this end Christ both died, and rose, and revived, that he might be Lord both of the dead and living.
1 Timothy 4:1-5
Now the Spirit speaketh expressly, that in the latter times some shall depart from the faith, giving heed to seducing spirits, and doctrines of devils; speaking lies in hypocrisy; having their conscience seared with a hot iron; forbidding to marry, and commanding to abstain from meats, which God hath created to be received with thanksgiving of them which believe and know the truth. For every creature of God is good, and nothing to be refused, if it be received with thanksgiving: for it is sanctified by the word of God and prayer.
1 Timothy 2:1-4
I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; for kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. For this is good and acceptable in the sight of God our Saviour; who will have all men to be saved, and to come unto the knowledge of the truth.
For this cause we also, since the day we heard it, do not cease to pray for you, and to desire that ye might be filled with the knowledge of his will in all wisdom and spiritual understanding; that ye might walk worthy of the Lord unto all pleasing, being fruitful in every good work, and increasing in the knowledge of God; strengthened with all might, according to his glorious power, unto all patience and longsuffering with joyfulness; giving thanks unto the Father, which hath made us meet to be partakers of the inheritance of the saints in light: who hath delivered us from the power of darkness, and hath translated us into the kingdom of his dear Son: in whom we have redemption through his blood, even the forgiveness of sins: who is the image of the invisible God, the firstborn of every creature: for by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers: all things were created by him, and for him: and he is before all things, and by him all things consist.
Be ye therefore followers of God, as dear children; and walk in love, as Christ also hath loved us, and hath given himself for us an offering and a sacrifice to God for a sweetsmelling savour.
But fornication, and all uncleanness, or covetousness, let it not be once named among you, as becometh saints; neither filthiness, nor foolish talking, nor jesting, which are not convenient: but rather giving of thanks. For this ye know, that no whoremonger, nor unclean person, nor covetous man, who is an idolater, hath any inheritance in the kingdom of Christ and of God.
Let no man deceive you with vain words: for because of these things cometh the wrath of God upon the children of disobedience. Be not ye therefore partakers with them. For ye were sometimes darkness, but now are ye light in the Lord: walk as children of light: (For the fruit of the Spirit is in all goodness and righteousness and truth;) proving what is acceptable unto the Lord.
And have no fellowship with the unfruitful works of darkness, but rather reprove them. For it is a shame even to speak of those things which are done of them in secret. But all things that are reproved are made manifest by the light: for whatsoever doth make manifest is light. Wherefore he saith, Awake thou that sleepest, and arise from the dead, and Christ shall give thee light.
See then that ye walk circumspectly, not as fools, but as wise, Redeeming the time, because the days are evil. Wherefore be ye not unwise, but understanding what the will of the Lord is. And be not drunk with wine, wherein is excess; but be filled with the Spirit; speaking to yourselves in psalms and hymns and spiritual songs, singing and making melody in your heart to the Lord; giving thanks always for all things unto God and the Father in the name of our Lord Jesus Christ; submitting yourselves one to another in the fear of God.
Be careful for nothing; but in every thing by prayer and supplication with thanksgiving let your requests be made known unto God. And the peace of God, which passeth all understanding, shall keep your hearts and minds through Christ Jesus.
And let the peace of God rule in your hearts, to the which also ye are called in one body; and be ye thankful. Let the word of Christ dwell in you richly in all wisdom; teaching and admonishing one another in psalms and hymns and spiritual songs, singing with grace in your hearts to the Lord. And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him.
1 Thessalonians 5:16-18
Rejoice evermore. Pray without ceasing. In every thing give thanks: for this is the will of God in Christ Jesus concerning you.
Wherefore Jesus also, that he might sanctify the people with his own blood, suffered without the gate. Let us go forth therefore unto him without the camp, bearing his reproach. For here have we no continuing city, but we seek one to come. By him therefore let us offer the sacrifice of praise to God continually, that is, the fruit of our lips giving thanks to his name. But to do good and to communicate forget not: for with such sacrifices God is well pleased.
As ye have therefore received Christ Jesus the Lord, so walk ye in him: rooted and built up in him, and stablished in the faith, as ye have been taught, abounding therein with thanksgiving.
Continue in prayer, and watch in the same with thanksgiving; withal praying also for us, that God would open unto us a door of utterance, to speak the mystery of Christ, for which I am also in bonds: that I may make it manifest, as I ought to speak.