Courting Miss Granger, about the Malfoy Marriage Curse. Said curse has been in force for several centuries, and dictates that the eldest male in every generation must marry a Witch — not necessarily pureblooded, but no more than one year older or younger than he is — before his twenty-sixth birthday, or the family fortune devolves onto a cadet branch and said eldest male will suffer a gruesome death. The Idiot Plot element comes into effect when NO ONE TELLS DRACO ABOUT THIS CURSE until his twenty-fifth birthday — when, conveniently enough, every Witch of his generation is conveniently married off ... except for one. And guess who that one might be?
Perfect Lionheart's Partially Kissed Hero definitely qualifies for this trope, as everyone but an Overpowered Harry has had their IQs driven through sub-basement levels to make any plotline in the story work.
The protagonists know that a girl will be raped by her stepfather when she returns home for Christmas? Instead of just keeping her at Hogwarts, they teach her needlessly complex spells and ship her home, so she gets her clothes ripped off before saving herself. A four-year-old and an infant won't eat the food being given, while locked in a dungeon? Two of the girls magically enlarge their breasts to provide milk. Possibly one of the dumbest moments is when Emily agrees to a bet with a Muggle girl named Rosaline. The terms are that if Jamie doesn't win a race, Emily must provide sexual favors for Rosaline. Keep in mind that Rosaline is seventeen, while Emily is eleven. Instead of refusing the bet (Rosaline gives the terms before Emily agrees) or going to Hermione or Harry to deal with a clear case of sexual harassment, Emily continues to panic in private, vomiting several times, before finally telling someone about it.
Not to mention Hermione's kidnapping, which wouldn't have happened if Hermione herself and the rest of the Hogwarts faculty hadn't somehow managed to get themselves outwitted by Crabbe and Goyle.
The notorious Naruto fanfic For Your Eyes Only. The only way this AU society could exist is if everyone in it was insane.
The first case of Brendan Namron: Ace Attorney relies almost exclusively on Night Russ's stupid lies to drive the trial. The second case is much better, thankfully.
Eiga Sentai Scanranger has abundant examples, but probably the best is "There's Something About Toni." The villainess's plan is to have a hate-inducing monster attack the school's Valentine's Day dance, and have a student use pent-up energy to blow up the rangers at said dance. Some problems arise 1) She doesn't appear to take any steps to make sure what the student uses his energy to blow up is the rangers 2) She doesn't take any steps to insure that the student will even be at the dance 3) She continues on with her plan to use her monster at the dance, even though she earlier used it in a flashy attack that got the rangers' attention and allowed them to discover how to reverse the effects of its powers. This end up being okay for her because the rangers seemingly forget they have this information 4) The villainess was taken by surprise when she found out another superhero would be performing at the dance, even though this was advertised on the dance's posters 5) When the monster shoots an arrow at the guest superhero, one of the rangers tries to perform a body block to save her. Even though the ranger in question's signature weapon is a shield, and moreover a shield she's supposed to be able to throw with Captain America-like skill. 6) Not really part of the Idiot Plot but still worthy of mention, the rangers waited until after classes to do anything about the monster's rampaging victims during its first appearance. Meaning also that classes were held during rioting on campus.
The notorious My Little Pony: Friendship Is MagicTroll FicPattycakes has this in spades. Sure, Dash. Just let your one chance to escape go rather than risk people seeing you in a nappy. Go ahead, Scoot, confront Psychoshy alone rather than get backup. Oh yeah, and everyone in Ponyville, feel free to indulge Fluttershy's growing obsession with mindraping everyone into submissionnote and this is the good interpretation - the alternative is that they're very much aware what they're doing and are simply evil.
Other notorious MLP Troll Fic Cupcakes is a milder example. So Pinkie is kidnapping and killing ponies to be baked into various foods. Disgusting, but fair enough. Nopony really notices the missing ponies or contacts the police. Okay, that's a little worse, but that could be hand-waved by the bystander effect. Pinkie is doing all her torturing and killing in the basement of Sugarcube Corner, the local bakery. Er, do the owners Mr. and Mrs. Cake ever go down into their own basement just once and notice that maybe the party streamers made out of dried intestines would be a bit strange?
An author-proclaimed (but non intentionally) So Bad, It's Good AU fanfic based on Pokémon and Sonic Sat AM called From an Ordinary Life to Extraordinary Strife features both the villains and the heroes acting like complete idiots, and the entire story wouldn't have happened at all if the authorities had paid any attention to the fact that a man just upped and created a new city and whoever moved there was never seen again.
In the seventh episode of My Brave Pony: Star Fleet Magic II, a robot attacks a dam. Nopony can figure out it's not a living being despite noticing it's made of metal and shoots Eye Beams until its wiring is exposed.
This sums up The Opposite Effect, an Ace Attorney fanfiction in which Pearl and Trucy decide to hook up Ema and Klavier, despite Ema clearly hating him. Somehow, the two repeatedly miss Ema throwing away Klavier's gifts to her and screaming abuse at him, while Ema never thinks to directly confront the two about their actions or ask Phoenix to do something about it. When Apollo finally gets around to asking them what they're doing and telling them that Ema doesn't like Klavier, they somehow get it in their heads that he's jealous and proceed to try to set him up with Trucy's history teacher and one of her classmates. At this point, still nobody talks to Phoenix about what's going on (despite him being Pearl's friend and Trucy's legal guardian). Meanwhile, Klavier spends the fic convinced that Ema loves him based solely on Pearl and Trucy's word, despite Ema repeatedly yelling at him, throwing his gifts out the window, and hitting him. In the end, the only way Apollo and Ema can think to resolve the issue is to trick Klavier into being set up with Trucy's classmate, essentially resorting to the same stupid tactic they were upset the girls were doing.