The ending of Mickey's Good Deed. Earlier in the short, Mickey sold Pluto to a rich man's Spoiled Brat son just so he could afford to eat. Even after spending the money to buy Christmas presents for a poor mother and her large litter of kittens (the eponymous good deed), Mickey sits by a campfire with a roast chicken, and laments that he'll never see Pluto again. Thankfully, Pluto escapes and rejoins his old companion in a very heartfelt scene.
Elmer Elephant, when the other animals are taunting Elmer for his nose and send him sulking away crying. Anyone who was bullied as a kid would remember how much that kind of treatment hurt.
Puppy Love in which Mickey and Minnie get into a huge fight when she thinks Mickey put a bone in her box of chocolates (it was really Pluto and Fifi) and they nearly break up.
"The Moose Hunt". Mickey grieving after he thinks he shot Pluto (who is playing dead).
Similarly, "Squatter's Rights," has the ending scene where Mickey thinks Pluto was shot, when in truth the butt of the gun knocked him out and Chip and Dale poured ketchup on him. Mickey has no idea of any of this though and thinks his dog was just killed, proceeding to cradle Pluto's head while tears pour out his eyes.
The ending of "Donald's Happy Birthday". Donald, thinking the nephews bought a box of cigars for smoke, he forces them to smoke the entire box in an almost sadistic manner. When he finds out that the box was actually a birthday present for him, he feels so guilty he shrinks to the size of a bug.
The featurette "Goliath II", where Goliath is spanked and branded a traitor for trying to run away.
The end of "Goofy and Wilbur." Wilbur was eaten by a frog and then the frog was eaten by a stork. Goofy is horrified and starts crying, but then attempts to cheer himself up, reminding himself that Wilbur wouldn't want him to do nothing but mourn and that he can find other grasshoppers. Then he breaks down even harder, yelling "But they're not Wilbur!" Goofy's crying isn't played comically over-the-top either: it sounds like someone genuinely grieving for a pet they loved.