As a watcher of The Simpsons from day one... it's really hard for me to pinpoint a favorite episode, much less a favorite moment, just because there's been so many good ones.
I'd say one of the most emotional ones (and yet very funny at parts) is probably the episode "One Fish, Two Fish, Blowfish, Blue Fish" in season 2, where Homer eats a deadly fugu (blowfish) at a local sushi restaurant and is told that he only has 24 hours left to live.
Needless to say, there's a happy ending, but the emotions leading up to that point are very powerful for a cartoon.