Borderline in The Life and Times of Juniper Lee. As the Te Xuan Ze, June has extraordinary abilities... and is physically incapable of leaving her hometown, meaning that all of her life's dreams are out of reach. Fortunately, the show doesn't harp on this fact too much, preventing Wangst.
Both Terra and Raven of Teen Titans have incredible kinetic abilities. The downside of this is that the actual control of each character's power is directly tied to their current emotional state, so when rage or panic overtakes them, chaos inevitably erupts. This was such a large problem for Terra that she eventually opted for high school instead of heroism.
It gets worse for Raven... when she loses control, she's liable to turn into a reasonable simulation of an Eldritch Abomination (just ask Dr. Light, if you can get him to put his experience in words). And, oh yeah, she was born to be a tool through which her demon lord father can kill everyone on Earth, then take it over. Luckily, she gets better after that.
This trope seems to be the main reason Jinx picked villainyat first; with her powers of bad luck, at least as a villain she could be respected by her peers for destroying things. Considering the sheer, mindboggling amounts of collateral damage the Titans can do (Cyborg using a building as an improvised weapon, anyone?)...
To a lesser extent, Starfire's powers are also emotionally based and thus can fail to work at unfortunate times. There's been at least one instance where relationship drama has caused her to lose her ability to fly. While she was flying at a high altitude.
And Red Star, from the episode "Snowblind", who's basically the Hulk, but a nuclear bomb instead of a rage monster.
The Spectacular Spider-Man's Electro is amazingly powerful, but suffers from extreme Power Incontinence, particularly when angry, rendering electrical devices inoperable or destroyed by power surges. He has to wear a suspiciously ineffective insulation suit at all times to avoid harming others, and can't even eat without discharging electricity. Small wonder that he freaksout and goes on a rampage.
Similarly, Rhino has his suit, which has him become super strong and completely impenetrable... Which is too bad, considering that he needs to rehydrate every few minutes, otherwise he'll pass out.
Cheetara actually states that she considers her clairvoyant ability to be more of a curse than a blessing, due to the physical drain it puts on her body.
During Lion-O's Anointment Trials, Tigra is shown as having the ability to make others see what is not there. The strain is apparently so bad we never see him do this again.
The version of Metallo on Superman: The Animated Series gains super strength, immunity to disease, and Nigh-Invulnerability, but loses all his senses of taste, touch, and smell in the process. The constant detachment from reality gradually drives him insane.
In the 1990's X-Men cartoon Jean Grey was often unable to use her Psychic Powers. It was common to see her passing out after trying to use them, saying that "she can't do (whatever she is trying to do) because her powers aren't as strong as Dr. X's", probably an action from the writers to not let her be too powerful or the others would be rendered useless. Not including her Phoenix-embodiment form.
Hank McCoy's "Beastly" appearance often prevent most people from understanding his incredibly cultured and intellectual mind, as well as his good natured heart.
In Batman Beyond, one villain used a belt that induced intangibility to phase through matter, which, as an opportunistic reporter, he used to discover Batman's secret identity and thus gain fame. Unfortunately for him, the more he used it, the more it affected him, and, eventually, his body became intangible without the belt. So intangible, in fact, that he could not stand or hold onto things, and he eventually fell through the ground to presumably die of either a fiery death in the earth's mantle, or asphyxiation. Or not.
Blight gets this too. He is turned into a living radiation bomb, and his skin glows green, making him look like a neon skeleton! His very touch can gradually burn through stone! Oh, and he's a big wig in a major world corporation, good luck keeping that a secret.
And there's the Terrific Trio. While their powers are great, their DNA is slowly decaying codons at a time.
Given an unusual treatment in Batman Gotham Knight, in which an anime-looking Bruce Wayne tests out a personal force-shield as part of his Batman gear. When it deflects a ricocheted bullet into the body of a gang thug, putting the man in the hospital, Batman decides that the shield is Blessed with Suck and discontinues using it, rather than risk having deaths on his conscience.
According to Rudolph and Frosty's Christmas in July, Rudolph's glowing nose was caused by the magic of the northern lights and serves as a means to defeat the evil Winterbolt, though the glowing nose did cause him to have a crappy childhood... This turns out to be a feature rather than a bug.
Ace from the DCAU is one of the most tragic examples. Her incredible Psychic Powers accidentally drove her parents insane, she was treated as a government experiment her entire life, she was used by a madman, and to top it off, when her powers developed to the point of altering reality, they also caused her to have a fatal aneurysm.
Aang from Avatar: The Last Airbender. Sure, he's the Avatar, a godlike human being who can manipulate all four elements to balance out the world. Sure, he's the bridge between the Spirit World and the regular world. Yeah, okay, he's the spirit of the planet all rolled into one dude. His past lives worked out just fine being the Avatar. Unfortunately for Aang, he doesn't get to wait until his 16th birthday to be told that he's the Avatar. Nope, some grumpy old dudes tell him when he's twelve because they fear a war is soon to break out. They make plans to take Aang away from his guardian, so Aang runs away...and gets frozen inside an iceberg for 100 years while a war sweeps across the world while he's gone. Then when he gets out of the iceberg, he has to fix everything! By himself! With a bunch of scary guys trying to capture him! And he's 12.
Plus, the first thing he learns about the new world he's entered is that he is the only Air Nomad left since all of the others were wiped out by the Fire Nation in the opening moves of the war, who were the priority targets due to having the Avatar.
Additionally, whereas every other Avatar (that we know of) had years to learn how to bend all the elements (Roku mentioned taking roughly a decade), Aang has less than one year.
Oh, and even after he can control the Avatar State there's a downside (for the world, at least): if he dies while in it, the Avatar Cycle is broken. No more Avatar. And given he's the spirit of the world and that killing the spirit of the moon made the moon disappear until it was resurrected...
The Firebending used during the war had its own suck. Being powered by rage, it can only destroy (as Jeong Jeong complained) and is more difficult to control, and make the Bender much more violent. And if you lose your rage, you lose your powers. Thankfully, the original form powered by willpower and joy of life offsets most of these problems.
The backstory revealed in The Legend of Korra paints the Avatar cycle as less of a blessing and more of a curse. It's basically a neverending attempt to atone for the first Avatar's mistakes which brought imbalance and strife to the world. Each Avatar is doomed to shoulder this burden, and none of them were able to create a lasting peace.
Danny Phantom and Vlad Masters have this. Unlike Danny, Vlad used it in rather unethical ways. Unfortunately for Danny, his parents want to kill his alter ego.
The Venture Bros. has Team Impossible, a very dark parody of The Fantastic Four. While Dr. Impossible has Reed Richard's stretching powers his wife Sally has invisible skin (making the flesh underneath visible), her brother Cody is combustible with oxygen and Ned is heavily callused (and retarded). Richard simply couldn't care less, being that he takes For Science! to the point that other people could matter less to him.
One Static Shock villain gained the power to absorb non-living matter in lieu of eating regular food. Unfortunately, the mass just accumulates (without a regular digestion system, he body wastes nothing) without actually increasing his size, and he eventually becomes so dense that he's incapable of moving.
And while Static's powers are usually just fine, there's one episode where sunspots mess with his powers and make them particularly strong. First it's kind of cool, but then he finds that he can't control them and the extra electricity is causing problems. Then they go very weak. Then a bad guy needs a beat down.
Time Zone from Static Shock. Her powers caused her to randomly slip back and forth through time. Gear built a belt that stoppped the random travelling, and enabled her to mechanically control her powers -with a remote control. Time Zone was then at the mercy of whoever had the remote.
Mystique Sonia from Hero 108 is cursed so that anyone who tells her they love her three times gets turned into a small hat-like creature called a Yaksha. While it comes in handy at times, it also means she can't really have a love life as no one can tell her they love her back.
In ReBoot, Bob's fusion with Glitch turns into this during My Two Bob's. His powers become a liability when they aren't needed anymore, are hazardous to his health, and lower his chances at winning Dot's heart over the "normal" Bob. This is because Glitch had been severely damaged when they fused together. There probably wouldn't have been any risk to his health if Glitch had also been healthy. Then again, if Glitch had been fully functional, the fusion wouldn't have been necessary in the first place.
In Gargoyles, it's eventually revealed that Owen Burnett is Puck in disguise. When Oberon decides to allow Puck to stay in the human world to be Alexander Xanatos's magic teacher, he slaps on the stipulation that Puck can ONLY resume his true form when he's teaching Alex—otherwise, Puck's stuck being a human.
On South Park,Captain Hindsight claims that his power of retrospect works this way—anytime something bad happens, he's cursed to realize how it could have turned out better. (His revelations are never particularly brilliant, like "they should have called the police earlier.") This also causes him to become incredibly paranoid, constantly question if the choice he makes is the right one or he will instantly know what he should have done. Mysterion (who is really Kenny in disguise) then retorts that his superpower (which is continually coming back from the dead, with everybody forgetting his death every time is far worse.
Think being the leader of a town and having everyone in said town admire you would be utterly awesome? Try telling that to Jack Skellington, who is the Pumpkin King of Halloween Town. However, it's always about Halloween, every single day, which is problematic when you want something different. Taking a break is not an option, as shown when Jack disappears for only two days, everyone is in a panic. Also, Jack seems to not be able to give the crown to anyone else if he wanted ("But who here would ever understand/That the Pumpkin King with the skeleton grin/Would tire of his crown, if they only understood/He'd give it all up if he only could . . . "), so he's stuck with the job. And if the scenes after "This Is Halloween" is any indication, due to being the biggest in-universe celebrity, Jack can't even having a decent conversation with anyone. And who knows how long Jack's been doing the Halloween job? He could be Really 700 Years Old for all we know. When you think about it, you can't blame the guy for desperately wanting to try out Christmas.
But because he's still the Pumpkin King, he can't escape Halloween even when he's trying to be Santa Claus. The result? Some very dangerous toys.
In one episode of The Smurfs, Smurfette is given a Midas touch power that enables her to turn whatever she touches into candy, but as in the case of King Midas turning his daughter into gold, Smurfette ends up turning Brainy into a Smurf sculpture of candy.
Rogue gets this trope as her power is to siphon off the powers, memories, life force, etc. of anyone with whom she makes skin-to-skin contact. Her downside is the same as the classic "Midas" example: can't shut it off.
Jean Gray is a remarkably powerful telepath, but in one episode, her powers get out of hand and she's overwhelmed by hearing everyone's thoughts and she can't control her telekinetic bursts.
Scott has his Eye Beams that are very useful as a crime fighter, but he's forced to wear his visor or glasses at all times. If he loses them (which tends to happen a lot), he has to keep his eyes closed, depriving himself of sight, or end up destroying everything in his path so he can see.
Blob and Toad are outcasts in school, thanks to their physical appearances that are tied into their powers.
Wanda's powers are so powerful that she was locked in a mental institution by her father to prevent her from destroying everything, though when she gets out it seems like she can control them just fine.
Kurt not only has his "Fuzzy Elf" real form (while it is cute, it freaks people out) - with that he can't touch people without them realizing he's uh fuzzy and two-fingered. It's revealed that his teleportation shunts him through what's best described as Hell to get from Point A to Point B and the one time they worked something to slow him down so they can figure how it actually works, the residents decided he looks tasty.
Every mutant is Blessed with Suck, in that once everyone finds out about mutants, just having special powers makes you instantly hated and despised by everyone.
The Morlocks are mutants whose powers give them non-human features.
Inspector Gadget. Think it over. The man cannot have any organic arms or legs anymore. He touches people with cold, cold steel. Something absolutely terrible had to be done to his head to accommodate all those gadgets, unless we allow for Hammer Space. He's either a mentally incompetent individual who was in no way right to give consent to have all those things attached to his body, or, something happened before or during his upgrades that turned him into the lovable moron he is in the series. And his gadgets fail spectacularly half the time.
And in a more recent fan mail episode, Peter gains the ability to turn people into Robin Williams by touching them with his hands. While delighted at first, he quickly regrets having this power after waking up next to Robin Williams instead of Lois. The episode ends with a world full of Robin Williamses and an insane Peter who has chopped his own hands off.
Josh McGrath/Max Steel of Max Steel. While an accidental infusion of nanoprobes gives him abilities including super strength, super speed, invisibility, and the ability to change his appearance, he comes close to dying before resident genius Berto suggests experimentally dosing him with transphasic energy ("The Max Probes need transphasic energy to survive, and so does your son. I think."), and there was the chance that it would just kill him faster. Naturally, it's a success, but Josh/Max needs continued exposure to transphasic energy to survive (there are at least three instances where a lack of energy comes close to killing him) — and in a worst case scenario, if enough of the probes are badly damaged or destroyed, then Josh/Max will die.
One episode of My Little Pony: Friendship Is Magic introduced "Cutie Pox", a magical disease that causes whoever catches it to break out in cutie marks. Whoever has it gains the talent that goes with that cutie mark, but is also compelled to practice that talent whenever possible. Poor Applebloom catches it, and finds herself compulsively hula-hooping, spinning plates, tap-dancing, speaking French, sweeping chimneys, and more.
The Mask had two recurring characters, teenagers who intentionally exposed themselves to radiation to get super powers. One became a powerful shapeshifter akin to Clayface, while the other became...a fish. A fish that could talk, but a fish nonetheless. To top it all off, a later episode revealed that he couldn't even breath underwater.
Though it may not seems like it, Futurama's Phillip J. Fry is, in fact, the most important person in the universe. Thanks to a time travel paradox involving copulating with his grandmother and thus becoming his own grandfather in the process, Fry inadvertently was born without a "delta brain wave", making him immune to the intellect-draining effects of the evil Brain Spawn race. Which he fights in a grand total of two episodes. The rest of the series involves him suffering from being a socially crippled, brain-damaged Cloud Cuckoolander.
Wreck-It Ralph: "WHY DO I FIX EVERYTHING I TOUCH?!" Mr Fix It powers are cool and all until you're trying to break out of prison and your efforts are simply making the prison sturdier.
Raven Queen from Ever After High has impressive magical ability... unfortunately, it's only in dark magic and she really has no interest in becoming the next "Evil Queen."