It seems that the issue here may be due to differing metaphorical purposes of burning coals.
Coals can be used to burn and harm, but they can also be used to resharpen and temper as in a forge.
From looking at the context of the verse from Paul, it would seem that his usage of the quote would have the latter purpose of to temper or reforge. Earlier he says:
Romans 12:9 - "Love must be sincere. Hate what is evil; cling to what is good"
I should think that doing good to one's enemies for the mere purpose of actually doing them harm later wouldn't be very "sincere."
The author of Proverb's contextual meaning on this verse seems to be more likely the former (harm) than the latter (tempering). This is from his earlier verse:
Proverbs 25:15 - "Through patience a ruler can be persuaded, and a gentle tongue can break a bone."
"Breaking bone" seems to have more in common with doing damage to one's enemies than bringing them into a state of righteousness. In addition, the verse in Proverbs has the the part, "and the LORD will reward you." added on where Paul does not. This statement of contrast with the burning coals again makes me think that he was meaning the coals would undermine or harm one's enemies. Though, it is entirely possible that the two authors could have been saying the same thing.