You're enjoying your day Everything's going your way Then along comes Debbie Downer Always there to tell you 'bout a new disease A car accident or killer bees You'll beg her to spare you "Debbie, please!" But you can't stop Debbie Downer!
In one episode it's shown that his father is the same way so it might be a racial trait. Being from Uranus, they're the butt of every joke, alongside Uranus simply being very dark and gloomy. Contrast Rosie, who was from Mercury and very bright and sunny.
Ted, the completely unsuccessful, miserable, and suicidal (in a humorous way) lawyer from Scrubs, who serves as the show's Butt-Monkey. Even The Woobie Elliot casually acknowledges that she is nowhere near as miserable as Ted.
Fraggle Rock has Boober Fraggle; who is not only depressive and fearful, but is also an obsessive expert on diseases, superstitions, and laundry. This actually proves very useful to the other Fraggles, as Boober is always the first to sense impending disaster.
Howard Moon from The Mighty Boosh. He's self-described as "dark, fractured, broken, paranoid" and has always been the “Stop Having Fun” Guy in all of the show's incarnations. Somewhat justified as the The Eeyore, as he really is the Butt-Monkey of his entire universe. In The Power of The Crimp, usually cheerful Vince is depressed and starts rattling off bleak imagery, to which Howard responds, "Have you got my script?"
Fisher, one of the interns on Bones. Most of his lines involve death, misery, and dying and he states that he is jealous of the corpses they work with. After the timeskip between seasons 5 and 6 it is revealed that he spent a year in a mental hospital for being too depressed to do anything but sleep 20 hours a day. Even Cam goes out of her way to try to get him to smile.
The Reverand Giles Shawcross in the Midsomer Murders episode "The Sword of Guillaume".
This troper's looking at you Meredith Grey! At least for the first few seasons...she's toned it down since she's married Derek but who knows how Lexie's death will affect her sunny disposition
Toby Ziegler on The West Wing. His ex-wife even cites this as the reason why she doesn't want to get back together. He does not take it well.
Season four of Hell's Kitchen had Matt as a particularly creepy example, as he would look to be on the verge of tears when told off and only found happiness when others were suffering, which he reveled in with unabashed glee.
David Hyde Pierce's character on The Powers That Be. (John Forsyth was the titular U.S. Senator; Pierce was his son-in-law who was in the House of Representatives.) One memorably darkly funny bit had him reading Final Exit (a real-life book detailing various methods of suicide) with commentary ("tried it, tried that twice, like that's going to work,....")
Mr. "Snuffy" Snuffleupagus in his early Sesame Street appearances.
Veep: Ben, the White House Chief of Staff, is constantly depressed, worn down by his long years in politics, especially in his current position. This is best exemplified at the end of "Crate", where while everyone else is celebrating the news of the President's pending resignation and Selina taking over, he merely says "Wow. This must be what it feels like to be happy" with the same dull look and voice he always has.
While it doesn't always fit, on Third Watch Sully often qualifies. At one point when a character mentions a first kiss, he responds that his first kiss sneezed in his mouth.
Hannah from Please Like Me suffers from depression. She doesn't really like going out or socializing, and is generally quiet and sullen when she does.
House, M.D. is such an extreme case, he legitimately seems to believe he can't function if he's not deeply unhappy.