Have you ever heard a remark so spectacularly dumb that it just made you angry? Do you have a low tolerance for Cloudcuckoolanders or ditzes?
While some might respond with sarcasm, pity, or even despair, these characters just can't handle the dumbness around them. They seethe, they rant, they curse or even resort to violence.
When this trait kicks in it can be the last straw for a normally reasonable person or an intense enough irritation to be their Berserk Button - especially for the Only Sane Man or those Surrounded by Idiots. Of course, this reaction is sometimes totally justified when the act of foolishness is or has put people in danger.
The results of becoming Enraged By Idiocy can vary from a Dope Slap to a Cluster F-Bomb to acts of violence. As with any loss of temper, this usually becomes a weak point or dangerous distraction for the angry party - making it a handy tactic for anyone merely playing dumb.
Obviously a Sub-Trope of Berserk Button. Characters who are the Only Sane Man or Surrounded by Idiots tend to have this, but it is also one of the main Jerk Justifications. You can expect the Tsundere to have this characteristic, especially if she's around the Idiot Hero.
Anime and Manga
In One Piece, while Luffy's extreme stupidity is enough to make all of his friends lose their patience now and then, Nami is the only one to really flip out every time he says something really stupid (which is half of the time; the other half he just says something somewhat stupid). She gets equally angry at the other crew members when they act idiotic. Usopp is becoming similar to her though he seems more enraged by the inability to think normally than stupidity per se.
In Sangatsu no Lion, Rei becomes incensed when Yasui doesn't learn from his mistakes both inside professional shogi (choosing not to learn from his losses) and outside (opting to once again drink his loss away instead of spending his last Christmas with his daughter as a family and initially pretending to forget about the presents he bought). Implicitly blaming Rei for the latter is what sets it off.
Deadpool—consciously or not—uses this to great effect on his opponents, distracting them with babble until they lose focus.
In the Pink Panther film series, Clouseau's constant bungling actually drives his superior, Chief Inspector Dreyfus, to try and outright murder him.
Animorphs: Visser Three. He has little tolerance for incompetence, and more often than not, it ended with whoever was being stupid getting offed.
In The Leonard Regime, Nick demonstrates this whenever Brandon says or does something stupid.
In The Lord of the Rings, Peregrine ("Pippin") Took accidentally makes a potentially enemy-alerting noise in the Mines of Moria (namely, dropping a bucket with chains down a stone well with lots of reverberations). The usually calm and compassionate Gandalf's response:
"Fool of a Took! Throw yourself in next time, and rid us of your stupidity!"
Considering what happened next, it's hard not to sympathise with him.
Live Action TV
Blackadder: Later incarnations of Edmund Blackadder are usually quite resigned to being Surrounded by Idiots. Whenever he does snap, however, it's usually because of Baldrick, George, Percy or The Queen doing something profoundly stupid.
In the early days of Breaking Bad, Walter was prone to extensive ranting at Jesse's screw-ups.
Doctor Who: Luke Rattigan, from season 4 of NuWho. He's an Insufferable Genius who, among many things, hate it when people refer to his creation as the ATMOS System. "It's a tautology! It stands for Atmospheric Omission System so you're just saying Atmospheric Omission System System!" Note: that's not a tautology.
Tywin Lannister from Game of Thrones. He's perpetually unamused, given that his standards are high, has no tolerance for incompetence and Joffrey's rule has been a long parade of follies and disasters.
Uther in Merlin. Most of the time it just resulted in an angry tirade and Merlin getting sent to the stocks, but he wasn't above having someone imprisoned or killed if it went too far.
Tony Soprano of The Sopranos has a Hair-Trigger Temper in general, but the stupidity of his mooks and his Son ranks near the top of his list of triggers. In a later episode, he considers murdering Paulie for his annoying and dimwitted tendencies.
Xena: Warrior Princess appears to experience this while she, Gabrielle, Joxer and Callisto (who is currently a god) are on their way to defeat another mad god. Joxer falls off his horse (again), and Xena becomes enraged and kicks him out of the party. In reality, she needed to get him away from the rest of the party so that he could retrieve a god-killing weapon without Callisto noticing.
Wess from MOTHER 3. When his son Duster brought back the wrong artifact from Osohe Castle, Wess FLIPS OUT AND SMASHES IT ON THE GROUND, then comes with Duster to Osohe Castle to guide him so he gets the correct artifact this time. For the rest of the game, he refers to Duster as "Moron."
Black Mage of 8-Bit Theater. As with most of life's problems, he responds to it with extreme violence.
Hank Hill on King of the Hill is this rather frequently. He even said as much on an episode where he had to go to anger management classes to end a restraining order filed on him by Dale, saying that he didn't have an anger problem, but an "idiot problem".