Fievel looking at a classroom through a window, especially since it was based off a story about Steven Spielberg's grandfather (also named Fievel), who couldn't go into a school because he was Jew and could only hear the lessons outside in the snow.
As a child (and adult) I could never get past Fievel calling out “Mama! Papa!” from the bottle without seriously sobbing.
In Fievel Goes West, when the cats attacked the Mousewitzs and the Toponis does it when you realize that these families had accomplished so much in the 1st movie, but they're still being attacked by cats. What made it worse was that Tony and Bridget now have a baby that was being attacked.
While The Treasure of Manhattan Island doesn't really have any specific moments that will leave you bawling, it's a surprisingly depressing movie in it's own right. The underground Native Americans decide that Europeans are incapable of sharing the upper world with them (having witnessed Police Brutality and a race riot), so they seal off the main tunnel. And the villains behind it only get off with a slap on the wrist.
Watching Fievel tear up in one scene after becoming disillusioned with America does tug at the heartstrings.