The "Ballad of Magellan" - not only for its morbid tone and ending, but there is something indescribably melancholy-sounding about the chorus, as if you can hear in their voices the Warners were aware their show was doomed (this was in one of the later episodes).
"Taming Of The Screwy" has Scratchinsniff reluctantly sending the Warners to the tower from a fancy party, even though he successfully de-zanitized them for the occasion.
"Meatballs or Consequences", when you think about it. The Grim Reaper comes to take Wakko away, and the other two decide that they want to go, too, rather than be separated from their brother. Subverted somewhat in that they're all awfully chipper about it.
Yakko: "Oh please don't separate us Mr. Death. We love each other. We're a family, a set, like Civil War chess pieces from the Franklin Mint."
In "A Christmas Plotz", the Ghost of Christmas Future (Yakko) shows Mr. Plotz his future: as a consequence coming out of Plotz's firing of Ralph, his son now is in charge, and Plotz is now the security guard. The Tear Jerker comes when Plotz himself is fired for not being able to catch the Warners; he just walks away with the most hopelessly depressing look on his face all while a sad version of "God Rest Ye Merry Gentlemen" plays in the background. Sure, you may not like him, but dang.
What makes it worse is that Plotz appears (to this Troper at least) to be undergoing Alzheimer's, meaning that Ralph's son has pretty much condemned him to death.
And the way that Thaddeus reaches out to his Alternate future self... (And personally for some strange reason I've always sort of liked him).
Rita and Runt's segments
"Smitten with Kittens". Rita finds three stray kittens and has to give them up so they can have a better life.
The whole damn opening at the pound is sad, but especially Rita's self-centered former owner complaining that she's too independent, won't come when she calls or comfort her when she's sad. Rita says "have a kid, lady," with a voice so weary you just know this has happened before.
Quite a few songs Rita sings fall under this. Particularly A Place Called Home and I'm Nobody's Mama. Ok, it's a cartoon cat singing...So why do I get all misty-eyed every time I hear these songs?
Maybe because her voice actor is a really good singer.
From the Les Miserables parodying episode Les Miseranimals. Rita sings A Flat In Gay Paree clearly modelled after the already tearjerking song Castle in the Clouds. It's really hard not to start tearing-up listening to Rita's longing for a loving home...especially given the horrid conditions she's in during that episode (She's imprisoned in a place where the owner, having run out of any other meat, has decided to kill cats and serve them for dinner).
"Puttin' on the Blitz", set in Poland during World War II, where Rita and Runt helps fend off the Nazis so that a little girl can catch a train to France with her resistance-leader father. Rita manages to get onto the train with the girl and her father, and she watches Runt getting smaller and smaller...before she shrugs and says to herself "Well, it was nice while it lasted," and goes back to beat the crap out of the Nazi wiener dog attacking Runt.
"A Gift of Gold". A piece of shiny gold wrapping paper is used to wrap Mindy's present. Needless to say, she tears it to shreds, but one piece blows out the window. It flits from place to place, trying to find somewhere to settle, before finally landing in a trash can. Just then, a poor man fishes it out of the can, using it to wrap a Cymbal-Banging Monkey, which he gives to his son. The son opens it, then folds the paper carefully and puts it in his pocket, giving it the love it sought the entire cartoon. That alone is heartmelting, but the cartoon is also voiced by Jim Cummings doing his impression of Sterling Holloway, i.e. "the most emotion-manipulating voice in the world"...
The Skippy & Slappy Squirrel episode "One Flew Over the Cuckoo Clock".
Skippy: I'll love you forever, Aunt Slappy.
Skippy yelling for Slappy, though fully aware that she can't come save him, as the social services agent relocates him to a new family.
Poor Chicken Boo, he should get at least one mention in this page, poor poor Chicken Boo.
The ending to ANY, if not ALL, of the Mindy & Buttons shorts.