The obvious answer here would seem to be Hamlet, any one of the seven versions I own, right? I mean, it's a tragedy after all. And I do generally cry at the end. But it leaves me feeling satisfied and cleansed, not sad -- it's really a cathartic play for me.
No, the absolutely saddest movie I have ever seen is The Fox and the Hound (1981). I have only seen it once all the way through, when I was in college, and I bawled the entire time. Even more than I cried the first time I saw Moulin Rouge! Buckets of tears.
When I was a kid, I had lots and lots of little Disney record-and-storybook sets. I loved most of them, and learned how to work a turntable by the age of 3 so I could listen to them without needing a parent around. I loved most of my records, except for three that scared me and one that made me cry. The latter, of course, was The Fox and the Hound. Tod and Copper swear that they'll "always be the best of friends," and then the evil grown-up world forces them to become enemies. If there's one theme that is guaranteed to make me sad and angry, it's enforced growing-up, and since that's the point of this whole movie, I pretty much hate it.