Angel Densetsu raises the Idiot Plot to an art form - the entire concept is that the main character unwittingly becomes the most fearsome gangster in Japan because of the whole cast's inability to communicate properly.
The ending of Gundam SEED Destiny hinged on the Minerva crew acting like complete idiots so that they would fight the Three Ship Alliance. The crew ignores Gilbert Durandal creating a fake Lacus Clyne and his plan to control people's destiny. They even let slide the fact that Durandal is willing to use his newly acquired doomsday cannon to blow away entire countries if they don't agree to his Destiny Plan. Actually between all of the characters the idiocy extends to most of the series.
Pretty much the entire second half of the Cardcaptor Sakura anime. It's painfully obvious to the main characters that Eriol is the cause of all the weird happenings, but no one is even the least bit suspicious of him. Most glaring is when Eriol buys a huge bag of sewing threat, cryptically says "There are many uses for thread" and wanders off. About five minutes later Shaoran is turned into a People Puppet and forced to attack Sakura with, you guessed it, thread.
Alex "Definitely Not A Villain" Brisbane: Step on the map. Yugi: Make me. Alex Brisbane: Oh, come on. I'll be your friend. Yugi: Look. There's no way I'm stepping on any freaking map. Alex Brisbane: What if I told you there was candy inside the map? Yugi: You've got to be kidding me. I'd have to be an idiot to fall for that. Tea: Candy? That sounds pretty good. Tristan: Yes. Let's go get the candy!
Sonic X takes this to absurd proportions when everyone on the planetFails Astronomy Forever for a couple of episodes, except Sonic himself, amusingly enough. Eggman rigs the moon to block out the sun, making it look like the moon pretty much stopped, along with the sun. Nobody finds this fishy because they've forgotten that the moon revolves around the Earth, and the Earth around the sun, not to mention the Earth spins on an embarrassingly wonky axis. Eggman takes the opportunity to try and brainwash everyone with the lamps he sells, only to have his plan foiled by Sonic yet again, because he was the only one not holding on to the gigantic Idiot Ball. The ball in this case being more of an Idiot Moon.
In X1999The Movie, the seven Dragons of Heaven each guard a magical barrier which serves to prevent The End of the World as We Know It. When the Dragon of Heaven protecting a given barrier dies, the barrier collapses catastrophically, and if all of the barriers are destroyed, humanity will be wiped out. So naturally, almost to a man, the Dragons of Heaven sacrifice themselves to kill the Dragons of Earth... who are trying to destroy the barriers to wipe out humanity. (The manga and the TV series handle things much better.)
In the anime of Sakurasou no Pet na Kanojo — the protagonist is specifically oblivious and thickheaded enough to be continuously embarrassed and ashamed of his female-lead just so hilarity may ensue. By a matter of fact, In the end of the first episode, the protagonist came to a life-changing, self-aware conclusion that he is merely a cog within the machine, a simpleton that is too afraid to be important; and that conclusion — is completely forgotten in the next episode; making it possible to repeat the same mistakes throughout the show, looking for the exact answer he had already answered on the first episode through the many hijinks which ensues with it.
Post exposition, it is explicitly shown that the protagonist has dealt with the tough, socially-inept, probably autistic heroine for almost a whole month; yet he is still shaken by her quirky, eccentric, abnormal behavior as if he just met her.
In Sora Wo Kakeru Shoujo, Nami's sisters structurally fail to acknowledge her obvious depressed state, going so far as to kick her down even more, as Kazane does in one episode. Even after Nami's Face-Heel Turn, Akiha simply tells Nami to shut up instead of listening to her during their first critical confrontation. Yeah, that really worked, now did it.
.hack//Legend of the Twilight runs on this to an absolutely unbelievable degree to the point that its predecessor .hack//SIGN is genius by comparison. At least in Sign, the main character Tsukasa is trapped in the established game, and can't leave until he finishes despite how dangerous it is. In Legend however, the main characters learn in the first episode that the game they're in is glitched and dangerous, and unlike Tsukasa they have the ability to leave whenever they want to. So what do they naturally do? They continue to play the dangerous game simply because they can. This is exactly why the series was criticized so heavily at this point.
Yuria doesn't want to have sex with Shunsuke because if she does she'll become his life-long sex slave, being a Sex Bot and all. She's pretty sure she doesn't like that idea, even after she starts wondering if she's fallen in love with Shunsuke. That's all well and good, but Shunsuke also doesn't want to have sex with Yuria because he is bafflingly devoted to his chaste, ice-cold girlfriend who he rarely hears from and even more rarely gets to see. And anyway what kind of idiot makes a Sex Bot who is intelligent and emotional enough to decide they don't want to be a Sex Bot? (oh right, that kind).
Simply put, the protagonist has a powerful demon sealed in his balls and can't ejaculate for a month or humanity will be destroyed. He does away with his Porn Stash, yes, and manages to deal with the various Horny Devils going after him, but doesn't stay away from his nymphomaniac-when-drunk coworker.
Which gets even funnier because he does leave her. AFTER the world is safe.
One of the two major conflicts in the final arc of Phantom of Inferno relied entirely on one of the characters coming to a braindead misunderstanding and the protagonist not only not making any attempts to correct her, but saying whatever he could to make himself look guilty and egg on the conflict.
Suzuka would be about a dozen volumes shorter if Yamato didn't find a way to say or do exactly the wrong thing every single chapter.
And on a much larger scale, the only reason there's any plot at all is because the Big Bad apparently never considered personally capturing Naruto before he became powerful enough to challenge him. Given his teleportation powers, it would have been a simple matter for him to snatch Naruto whenever he wanted, but he doesn't actually try until after Naruto gained full control of the Kyuubi's power, and is actually capable of putting up a fight. It wouldn't even have been hard, just pick any of the dozens of times before he got Jiraiya's supervision where he was either on his own, or only with other low level ninja and snatch him. It's not even in his own home village either, it's his JOB to routinely leave the village on little missions with only one mentor protecting him.
Here's your obligatory Hand Wave: It was because Itachi Uchiha was in his way the whole time. Itachi joined Akatsuki to ensure it would not lay a finger on Sasuke or the Leaf. He would have intervened if Tobi tried to attack Kohona in ANY way, shape, or form. That includes going after Naruto. Up until Itachi died, Tobi put on an Obfuscating Stupidity act. When he kicked the bucket, Tobi made his move. However that doesn't explain why he didn't do it before then.
A possible explanation there verges on Voodoo Shark. Every other jinchuuriki was successfully captured without significant resistance by their village because none of them cared that their walking weapons of mass destruction were being stolen by an organization of criminals. The reason Naruto and B were left for last was simply because they were the only ones their respective Kages actually cared about.
The canon explanation is that the Kyuubi has to be sealed last, so it makes sense to leave it last- though, admittedly, that does run the risk of losing the Kyuubi if Naruto is fighting somewhere else and dies (the Kyuubi will eventually be reborn, but after an indeterminate time). It is also established that Tobi wanted Naruto and Sasuke to fight each other first (for reasons unknown, but possibly related to his obsessions- since he is Obito, Naruto reminded him of his old self, so he wanted Sasuke to beat him to prove his old way of thinking wrong...as a wild guess). Its later established that he doesn't need the Bijuu per say, just a certain amount of their chakra, though the full Bijuu is preferred. He manages to get both without capturing either B or Naruto, because sources of their Bijuu's chakra was elsewhere (in a lost limb of the 8-tails and with two deceased villains for the 9-tails, respectively).
The Blood Prison movie, Naruto is sent to prison in order destroy the villain Artifact. However, Naruto was never told about any of this.
"Stink Bomb", from the anime anthology movie Memories is about a salaryman at a pharmaceutical company who takes some experimental pills - thinking they're cold medicine - and becomes the vector for a chemical agent that kills everyone within several hundred meters from him. From then on, everyone just gets dumber. His immediate superiors call him out of a quarantine zone to deliver the pills to them. The JSDF try to kill him, but are so incompetent they just cause massive property damage and only exacerbate the problem. The American military screws things up even more by trying to capture him alive to get access to the chemicals. And throughout all this, the salaryman just can't figure out that he's the source of the obnoxious stink and no one EVER bothers to try explaining it to him. It's played for laughs, but galling all the same.
In Lost+Brain the entire plot is basically one when Fridge Logic kicks in. The plot is based off Hiyama's plan of using hypnosis to remove human weakness with him controlling them having no chance at working, which any amount of additional research would have told him. The plot get even more stupid with the fact that Kuonji and the police never seems to even try to investigate Hiyama even though all the trouble happening can pretty much be linked to him.
Little Jumper would be a lot shorter if Chimari had thought to look up her mother's name before going back in time. But then again, the Idiot Ballis an Ichinose family heirloom.
Infinite Ryvius would have been resolved if they had transmitted what happened to the rest of civilization, and definitely would have if they simply landed or had somebody leave the ship.
In Zeta Gundam a lot of trouble would be avoided if the Argama crew could control the catapult deck traffic better.
The very first arc in CLANNAD hinged on the entire cast failing to have the common sense to question or search out a reason for why one character was being unseen and forgotten by more and more of the school and town. One member decides to investigate alone, and then immediately forgets and becomes unable to see said character. After that, they explicitly avoid this. It's not enough.
Any pre-Unova Pokémon episode is guilty of this, as Team Rocket would show up in every episode wearing easily transparent disguises and the protagonist would always fall for it, then act surprised when Team Rocket tries to escape with their Pokemon. Post-Unova sees the light of day dawn on a loads more competent version of the Team Rocket Trio.
Guardian Force, the latter half of Zoids: Chaotic Century, features a truly spectacular idiot plot in which Hiltz, The StarscreamBig Bad, wants to revive the Ultimate Death Saurer to Kill All Humans until only his race, the Ancient Zoidians, remain. In order to do so, he needs Fiona to open the door to Zoid Eve. The sensible thing to do would be to trick her into doing so at the beginning of the series, when the Ultimate Death Saurer would be unstoppable. Instead, he goes on a totally pointless tangent, building the Death Stinger and using it to go on a seven episode rampage, giving the main characters plenty of time to build the superweapons and develop the skills and experience needed to defeat him. Hiltz didn't need the Death Stinger for his plan; in lieu of the revelations made during the final four episodes, everything which happened before comes across as For the Evulz in-universe, and To Sell Toys out-of universe.
The entire short "break-up" arc in the Sailor Moon R anime could have been resolved by Mamoru telling Usagi the truth. Yes, he had prophetic dreams that she would die if they stayed a couple, but that did not require that he toss her to the curb and dump her. Then it gets resolved when Usagi finds out anyway and he completely caves. The real kicker? The dreams were being sent by the future King Endymion (Mamoru's future self) to try and strengthen the couple's love. That's right — he broke them up to make them love each other more.
In Fullmetal Alchemist, Envy is eventually reduced to a powerless form and May Chang has to return to her home country. Trouble is, the plot requires them both to be in Central for the climactic battles. Arakawa's solution is to have the older and wiser people around May put manipulative, sadistic Envy in a jar so that May can get the secret of immortality from him... the hideously nasty secret that they have all refused to use. What's worse, she's supposed to take him to the Emperor of Xing, a man who she said five episodes ago should never know about philosopher's stones because he would happily perform the required slaughter! Envy talks her into going back to Central in about five minutes.
In HunterxHunter, the Greed Island arc would have been just a little shorter had Gon swallowed his pride and just stuck to the damn plan, instead of try to take the Bomber head on just to prove himself. Even when he reaches a point in battle where Biske told him he needs to run for it, he's still too selfish to stop (even acknowledging it), and ends up losing a hand and having his right arm mangled for his trouble.
The Black Knights betrayal of Zero in Code Geass. Several years since the show has ended and it still is the most controversial element of the series. After Schneizel reveals that Zero is Lelouch, former prince of Britannia, and a wielder of Geass, they take his word for it with almost no argument. They then hold Zero/Lelouchat gun point, demanding a confession from him. While Schneizel was technically telling the truth, he was still their enemy, and no one bothered to question whether he was lying or manipulating them. They also don't figure out that their ability to mutiny proves that Zero had not, as Schniezel claimed, Geassed them into obedience.
Every single episode of the Island/Africa arc in Nadia: The Secret of Blue Water. There are so many holes in the "plots", that one could drive a truck through them.