The whole third segment of Cat's Eye View: Homeward Bound. Furrball's cardboard box house falls apart in the rain and he must seek shelter. He befriends a kitten and a rich girl, but the kitten's parents and the girl's parents don't want a filthy cat in the house. Tossed into the streets, he sees a homeless boy who's also not welcome in their house. So Furrball shares his lunch with him. They may not have the best life, but they now have each other.
Furrball and the ribbon in Strung-Along Kitty.
Heck, most Furrball centered episodes tended to be downers, given the poor little guy's miserable lot in life.
The newsclip from Fields of Honey, particularly when Bosko fades away into oblivion.
Babs:(crying) That's the saddest thing I've ever heard.
This is even worse when you realize the same thing will happen to the Tiny Toons, due to them being forgotten over time. Someone actually did a fanfic of this...it was depressing.
Later, when Babs realizes that the old lady she encountered turns out to be Honey, who turns back into her old youthful self when everyone laughs at her cartoons again.
The ending of "Washingtoon", where the evil Moral Guardian is reunited with her "tooniness", who she hasn't seen in years. This gives her a sense of humor and she bounces off happily to Wackyland.
Just before Buster and Babs go to appeal before Congress, they stop in at the Lincoln Memorial. Babs reads off a quote from the Gettysburg Address, when Buster admits that it doesn't apply to toons and that they were a "lost cause." What follows can be considered the greatest motivational moment of Western Animation.
Abraham Lincoln: "Lost causes are the only causes worth fighting for... Stand tough in the face of adversity, let right be done, let truth be your guide, and most of all, never give up. Never EVER give up."
It turns out to be Reagan talking to his doll, but his rambling still motivated them.
"Night of the Living Pets" has a Downer Beginning with Elmyra looking through a photo album of all the (good?) times she had with her now-deceased pets. Even though she was obviously the cause of their deaths, the poor thing looks like she was about to burst into tears any minute.