"They have laid Jerusalem in heaps" (79:1). When did that happen, and who is the Asaph of the title?

The psalm doesn't say when. Its title is only "A Psalm by Asaph" (79:1). The chapter on Psalm 50 sets out what else Scripture says of a man named Asaph, and doesn't say whether he is the one the titles name; Psalm 79 adds nothing to that. The psalm names no enemy beyond "the nations" (79:1), and no king and no date. Other books tell of Jerusalem's fall. 2 Kings says, "He burned the LORD’s house, the king’s house, and all the houses of Jerusalem" (2 Kings 25:9), and "All the army of the Chaldeans … broke down the walls around Jerusalem" (2 Kings 25:10). Lamentations says, "she has seen that the nations have entered into her sanctuary" (Lamentations 1:10), and in Hebrew its the nations have entered is the same two words, in the other order, as the psalm's the nations have come into (Psalm 79:1). And in Jeremiah 26 some of the elders (Jeremiah 26:17) recall Micah's words, "Jerusalem will become heaps" (Jeremiah 26:18). The Hebrew Bible has that word for heaps only four times: in Micah 1:6 (of Samaria) and Micah 3:12, in Jeremiah 26:18, and here. None of these books names this psalm, and the psalm names none of them, so these notes don't say it is about any one event. The chapter on Psalm 74 sets that psalm's burned sanctuary beside 2 Kings and Lamentations in the same way.