Playing The Road to Isengard and am on Stage 2B:

If I declare a side quest to be active for the given turn, but then make enough progress on a location to discard it, do I place damage on Side 2B of Breaking Isengard?

Initially I thought "no", but after re-reading the side quest rules I think that you still do.

Yeah, the quest card's text remains active even if it is not the current quest. When you explore a non-unique location with a side quest in play and selected as the current quest, you'll still place damage on the main quest card "Breaking Isengard."