Romans 12:20This was nothing new. In fact, it's a paraphrase of a passage from Proverbs:
Therefore if thine enemy hunger, feed him; if he thirst, give him drink: for in so doing thou shalt heap coals of fire on his head.
Proverbs 25:21-22The significance of this passage is related to the person receiving eternal consequences. It's not simply that they will feel guilty.
If thine enemy be hungry, give him bread to eat; and if he be thirsty, give him water to drink: for thou shalt heap coals of fire upon his head, and the Lord shall reward thee.
Coals of fire are a symbol of the wrath of God:
Ezekiel 10:2More particularly, they are a symbol of the wrath of God visited on those in hell:
And he spake unto the man clothed with linen, and said, Go in between the wheels, even under the cherub, and fill thine hand with coals of fire from between the cherubims, and scatter them over the city. And he went in in my sight.
Before him went the pestilence, and burning coals went forth at his feet.
Psalm 140:9-10Advocates of annihilation may be tempted to treat this as referring only to an interim punishment before the great judgment day. However, notice that this says that they will "rise not up again." The point is that they are stuck under these punishments forever.
As for the head of those that compass me about, let the mischief of their own lips cover them. Let burning coals fall upon them: let them be cast into the fire; into deep pits, that they rise not up again.
Furthermore, the point of the coals is not simply annihilation, but pain. What's the value of trying to "heap" coals of fire? It really seems to make little sense outside the context of conscious torment. While the eternality of that torment is not immediately apparent from Romans 12 itself, it becomes more clear when taken with the rest of Scripture's testimony.