Subverted when he won a large sum of money. He smiled from ear to ear, and leaped across the screen. This actually was how he ended several cartoons, bouncing and cheering madly, then returning to his sad state, looking at the camera and saying, "You know what? I'm happy."
This itself was subverted in the Harvey Birdman, Attorney at Law episode "Droopy Botox." Droopy sues his plastic surgeon for malpractice after his face freezes into a perpetual grin, and when he loses the case as a result of Harvey exhibiting Droopy's signed indemnity waiver in court, his weeping grin haunts Harvey throughout the episode.
Toot from Drawn Together is an example, though she is so bitter and repulsive that she really garners no sympathy whatsoever.
Happy Time Harry in the Aqua Teen Hunger Force episode "Dumber Dolls" takes this to comically dark extremes. He's a children's action figure who is depressed, addicted to pain pills, has a pile of unpaid "Action Bills" as his main accessory, occasionally practices self-harm with a knife that's permanently attached to one hand and routinely threatens to commit assault or suicide. In later appearances, he mentions he's on probation too (and missing a meeting with his P.O. by participating in the villain's Zany Scheme). Somewhat of a deconstruction. He shows you what kinds of things need to happen to a guy for him to have such a negative outlook on life. And by "things" I mean "Wife kicked him out, numerous health problems, overdue child support, overdue bills... this can go on. It certainly doesn't help that he took to the bottle to alleviate the pain he faces.
Mentioned or at least implied to be a cultural trait common to all Merbians. Terra Merb is so riddled with dangerous creatures, dangerous terrain, dangerous weather, and dangerous combinations of all three that Merbians have simply come to expect the worst out of everything, since it usually happens to them. It's so bad that when something nice happens to the Hawks, Stork can't enjoy it because he's too busy waiting for the punchline. Unfortunately, since Merbians developed their pessimism as a survival mechanism, he has a distressing habit of being right.
Another Hanna-Barbera property the same era was a 1968 reworking of Gullivers Travels. One of the Lilliputians who hung out with the hero was a hard-core Eeyore named Glum; for many, he was the most memorable thing about the show.
She veers strongly into The Woobie category, depending on how the writer treats her. There's a post cancellation episode where Peter acknowledges that she's not as bad as she deserves to be treated and promises to be nice to her... when nobody else is around. When Lois then enters, Peter proceeds to throw his drink in her face to keep in character.
It's worth noting that she occasionally borders on Jerkass Woobie; since everyone treats her like crap regardless, she eventually goes on a Only Sane Man rant about how she is essentially a victim of a variety of domestic abuse as everybody downright hates her for no reason. However, she reverts back to the Eeyore role again when she realizes being the Eeyore is pretty much keeping the family together.
Ironically, during one episode of the cartoon series, the entire cast notices Eeyore, just sitting on a hill and staring and try to cheer him up. The episode ends with Piglet asking Eeyore why he was so sad while Eeyore is still sitting on the hilltop, watching a beautiful sunset, and Eeyore simply asks "What makes you think I'm sad?"
Mr. Blossom of The Dreamstone is about the one resident of the Land of Dreams with a hint of cynicism, so much he seems to hold the entire village's worth.
Bill Dauterive from King of the Hill, who fell from being a handsome star linebacker to being a fat, balding, divorced sad sack. In particular the Christmas episode "Pretty Pretty Dresses" where he's fixating on his divorce and spends the first half of the episode reminding everyone of their own failed relationships.
In the My Little Pony: Friendship Is Magic episode "A Friend in Deed", Pinkie Pie is shocked when a new inhabitant moving to Ponyville, Cranky Doodle Donkey, is such a humourless, unhappy Deadpan Snarker that, unlike absolutely everyone else, he doesn't like her and doesn't want to be her friend. She spends the rest of the episode trying to defy this trope, which only makes things worse until she figures out the reason for his melancholy. He's spent years searching for his true love, whom he lost touch with due to a misunderstanding, and become more and more depressed until he's finally given up and decided to retire to Ponyville with his memories... But it just so happens that she lives there as well, and Pinkie Pie, of course, knows her. Needless to say, Cranky is cured of being The Eeyore by the end.
PJ on Goof Troop oscillates between this and a very compassionate variation of a Stepford Snarker. He can be happy sometimes, and the fact that he usually isn't is evidence of the injustices done to him, both immediately beforehand and throughout his entire life. He's somewhat better in the first movie, and in the second movie, his unhappiness is finally rectified.