I admit that sometimes I'm a sucker for a tearjerker, but it depends on my mood, the context, and whether it strikes me as being _intended_ to be a tearjerker.
The older ones, like "The Vacant Chair," "All Quiet Along the Potomac," "Katy Wells," and many other war-related songs, usually get to me because they have (I think) a more authentic emotion about them. Modern stuff like "Honey," "The Christmas Shoes," and (sorry) even "Teddy Bear" just strike me as too formulaic, too calculated, to generate real emotion.
Understated songs like Jim MacLean's "Don't Cry In Your Sleep" sometimes catch me by surprise... the real emotion behind the words, can be much more powerful than anything worn on the singer's sleeve. Another one I just can't sing, tho' I really like it, is "Mama Look Sharp" from the musical "1776."