Several characters from Batty Battalion qualify, but Rookie and Scout are most obvious, (Rookie verbally and Scout physically)
The title character in the Charlie the Unicorn videos. The blue and pink unicorns apparently come up with extremely convoluted and surreal plans just to make Charlie miserable. This is probably why Charlie is a Deadpan Snarker.
Strong Sad from Homestar Runner, whose two brothers Strong Mad and Strong Bad enjoy heaping physical and psychological abuse on him.
Also from Homestar Runner, almost everyone in Strong Bad's comic book Teen Girl Squad.
Nebulon, Rumble Red, the King of Town, Cartoon Mary Palaroncini, Coach Z, and 1-Up are also frequently abused. There's only one person who does like Nebulon's style... and she's an enormous alien cow. However, it's justified in Red, Mary's and 1-Up's cases, in that they're parodies of annoying cartoon sidekicks.
Homestar Runner himself can also qualify, considering how many times Strong Bad has insulted, assaulted and tricked Homestar for the sake of his own enjoyment. That and Marzipan keeps breaking up with him. Bubs also squeezes Homestar for all the money he can get, but Bubs does that to EVERYBODY.
Germaine in the Neurotically Yours cartoon series. She's a magnet to stalkers and perverted men, has trouble controlling her weight, is the butt of Foamy's criticisms and Jerkass rage, and can't get any of her poetry published. She recently shaved her hair and gained more weight as a desperate attempt to get men to stop hitting on her. It doesn't.
Washington sort of strays into the territory when you consider his speech at the end of Recreation, leading to a possible case of The Dog Bites Back:
Chairman: Agent Washington, when you find these blue soldiers that you're talking about, what makes you think that they are just going to give you the Epsilon unit when you ask for it?
Washington: Heh heh. For as long as I can remember, I've been lied to, taken advantage of, shot in the back and left for dead. And now, I have a way out of all of this. What in the hell makes you think that I'm going to ask for it?
Church is kinda the universe's Butt Monkey. He's been tortured so his mind fragments into pieces, killed more than once, had his girlfriend die more than once, trapped—forever, as far as he knows—in a virtual world, and just generally always been smacked down again every time he thinks life is starting to look up.
Ultra Fast Pony has a few. Spike is viewed by nearly everyone as a source of unpaid labor and physical humor. (It's lampshaded at one point with his line, "I swear, it's like everyone hates me or something!") Fluttershy is a former slave who continues to get exploited because she lacks a backbone. Rainbow Dash has never friends or family for any length of time—partly because of her obnoxious personality, and partly because she's blue. Both Fluttershy and Rainbow Dash know how unpopular they are: at separate points they each try to coerce Twilight by threatening to hang out with her. When Princess Luna isn't getting banished to the moon for no reason, she's forced to live in the castle basement where no one visits her, and her sister sends her letter bombs. Twilight starts off somewhat well-liked in Ponyville, then loses respect as the series goes on, until no one's willing to come to her birthday party. When Luna comes to Ponyville, Twilight is ecstatic that "Someone is even more hated in Ponyville than I am!"
Sage is the king of the butt monkeys. He has been shot in the same knee cap. twice. He has been shot with a tank twice once that was driven by his very drunk "friend". And then she proceeded to bury him alive. He was also shot again by his "friend" Zero when Venus gave the FIRE! comand. Before he was shot by yet another tank she'll he was shot by an unknown sniper in the knee cap...again. He has also been transported to another dimmention where he was knocked out and dragged to a torture basment.
With Sage being the king. Delta would be the prince. DeWitt was abandoned by his parents on a planet dubbed "Delta" where he was raised by a colony of tentacle monsters, at age eight he married his wife "Keeglor" and by age nineteen he ate his wife. At age twenty he was rescued. During season one he became a drug addict living under a bridge eating rats and snorting cocaine. His drugs are stolen from him and an Intersexual southern red neck promises she can recreate the sex he had with his old wife. In return he has to go to Tank and say the following.
"Delta:" Hey baby, you gonna keep all that hot spice to yourself or are you gonna give daddy a piece of that hot jalapeņo ass.
This did not end well. He wakes up from a coma after Tank skinned him alive and burned his body. Scoots and Dinky bring him back from the brink and force him to be their slave after a life debt. He becomes a sex slave to Scoots and a guinea pig to Dinky.
Jaune Arc from RWBY counts as a deconstruction of the so-called "lovable idiot". Initially, his battle incompetence is Played for Laughs, but during episode 12, it is revealed that his lack of combat savvy gets him bullied by Cardin, and it shows as he has absolutely no self esteem. He even revealed to Pyrrha that he'd actually stolen some transcripts to get into Beacon Academy and has had no experience fighting whatsoever. He'd been blackmailed into doing Cardin's essay along with his own in a very cramped schedule and is forced to stay up very late.
Many characters from The Big Lez Show qualify, but most noticeably Clarence, Quintin, and Norton. Norton is justified in that literally everyone in the series hates him.