Psalm 140:9-11 provides one possible answer, since there appears the same parallel of coals falling upon the head. Most English translations group verses 9-11 as one paragraph; the LXX and Masoretic Text (MT) group the entire psalm as one unit.
Psalm 140:9-11 (NASB)
9 As for the head of those who surround me,
May the mischief of their lips cover them.
10 “May burning coals fall upon them;
May they be cast into the fire,
Into deep pits from which they cannot rise.
11 “May a slanderer not be established in the earth;
May evil hunt the violent man speedily.
The LXX provides some slight nuance that the coals "fall upon ... the earth," which suggests to the reader that the source of the coals is heaven.
Psalm 140:9-11 (LXX)
9 [As for] the head of them that compass me,
the mischief of their lips shall cover them.
10 Coals of fire shall fall upon them on the earth;
and thou shalt cast them down in afflictions:
they shall not bear up [under them].
11 A talkative man shall not prosper on the earth:
evils shall hunt the unrighteous man to destruction.
The imprecatory prayer is that the "talkative" slanderers suffer the mischief of their own lips through divine intervention (and so to let them reap what they sowed to its maximum extent). That is, the imprecation is that from heaven burning coals would "precipitate" divine discipline, which is the imagery from the LXX. This backdrop from Psalm 140:9-11 therefore provides the perspective to understand Paul's words.
Romans 12:19-20 (NASB)
19 Never take your own revenge, beloved, but leave room for the wrath of God, for it is written, “Vengeance is Mine, I will repay,” says the Lord. 20 “But if your enemy is hungry, feed him, and if he is thirsty, give him a drink; for in so doing you will heap burning coals on his head.”
The conclusion is that when we are kind to the erring individual (instead of repaying evil-for-evil), we exclude any animosity. In fact, when we pray for our enemies we protect ourselves from bitterness. In this manner, the way is now wide open for the coals from heaven to come down upon the erring individual. In the context of Psalm 140:9-11, the burning coals of discipline are that these people would reap what they sow to its maximum extent.