As I've gotten older I've noticed more and more my emotions can be toyed with by films (possibly even more so since my son was born) and there are several that spring to mind.
Eternal Sunshine of the Spotless Mind
The Fountain
Me, Earl & the Dying Girl
The Butterfly Effect
Pretty much anything that involves a father/son relationship, especially if the father lets down his son in some way (clearly a huge fear of mine!) will set me off.
My little boy is starting young with it too, having already seen it a few times, on one watching of Big Hero Six, near the end (you'll know the bit I mean if you've seen it) he burst into tears and was pretty much inconsolable for about half an hour, seeing him so upset naturally made feel terrible and sad too, it was a right snotty and tearful mess between the two of us!! lol