The biggest one is Barber's Adagio for Strings. I happened to listen to that one on a day when I was feeling really guilty about something - and rightfully so - and it combined with my already depressed mood to make me feel like going somewhere and curling up sobbing in the fetal position.
As far as songs with lyrics go, the only things I can think of are some of the songs from Les Miserables, especially "On My Own" and "A Little Fall of Rain" as mentioned before. It didn't happen on my first exposure which was just a recording, but I happened to see some videos of those performed and that set some tears rolling. I can't even imagine what it would be like live, I'd probably cry like a baby. Buckley's "Hallelujah" doesn't quite move me to tears, but it does make the spine tingle a bit.