Okay, this is a weird one for me, probably because I don't listen or care about lyrics. And I can't think of a single song that moves me to tears on melody alone. I went through my iTunes listings searching and couldn't turn anything up. That's in the true "song" category. The closest would have to be my favorite Bobby Darin song, "I Will Wait for You." Now, in operas... there's quite a few arias that make me cry, however usually only when I'm listening to the full opera. And not just because the aria is sad, I often cry when it's beautiful too.
And there's quite a few orchestral soundtrack pieces that make me cry, usually Jerry Goldsmith (the scores to Lonely are the Brave, Masada, The Blue Max, The Edge, The Chairman are big culprits), but not always. Frank deVol gets me just about every time I listen to The Flight of the Phoenix score with "Gabriele's Death." It's a case where the music affects me far more than the actual scene in the film. The finale of George Bassman's Ride the High Country score usually gets me, but that one's definitely context-related. Miklos Rozsa makes me tear up with Ben-Hur and Ivanhoe. The Egyptian (Newman/Hermann). I think I cry far more in music because it's beautiful than because it's sad. I'm just a sucker for beauty.