Nobi Nobita from Doraemon could be considered the Ur-Example of this trope for modern anime. His father was athletic but book dumb, while his mother was frail but intelligent. Unfortunately he happened to inherit the worst traits from either parent, so he constantly gets left behind in both school work and physical activities. However, he has proven time and again that when he really wants to he can overcome his problems and really get things done.
Ash Ketchum from the Pokémon anime, though how idiotic varied over the years. He arguably got smarter in regards to being a Pokémon Trainer, but dumber when it came to everything else. It doesn't help that the anime frequently seems to reset Ash's intelligence upon entering a new region.
One supporting example of this is in the episode "Gotta Get a Gible!", where he and Pikachu attempt to make a jump from one side of a cliff to the other without noticing a bridge that Brock and Dawn use to make it across.
To summarize: Ash was a total idiot when he started his journey, and then gradually matured over the course of the Original Series and Advanced generation series. During the Diamond and Pearl series he gets even better at Pokémon battling, but, as mentioned above, he gets less intelligent in all other regards. During the Best Wishes series, however, he hits the Reset Button very hard and actually becomes MORE of an idiot than when he started out. During the XY series, however, he regains his former maturity and intelligence with interest, and almost averts this trope completely.
Similarly to Ash, considering they're based off the same character, Red, from Pokémon Special. He ends up maturing past this, but Gold and Black from later arcs pick up the mantle.
Black is a variation; he's actually incredibly book smart and studies a lot on his own time when it comes to Pokémon battling, but his (initially) self-centered attitude and poor social skills make him unaware when he inadvertently causes problems for other people.
In an SBS column, Oda replies to a question asking why Luffy is always so calm with "Probably because he's so stupid."
The Impel Down arc was partially a deconstruction of this trope, partially an attempt to evolve Luffy beyond it. Luffy's stubborn refusal to stand down or retreat from Magellan ends up almost getting him killed and costing him several years of his life. He attacks Blackbeard in spite of his considerable power, and the only things that keep him from getting into a fight that would be counterproductive to his mission are Jimbei restraining him and Blackbeard not looking for a fight. His pushing himself too far results in his body giving out, and Ace dying to protect him. Many of the strategies he'd fallen back on in the past (such as using brute strength to win and never backing down) fail as he comes up against many opponents he cannot defeat, thus forcing him to either do things in a more intelligent way or get even stronger.
Not a complete deconstruction, though. Luffy still has elements of an Idiot Hero after the events in Impel Down and Marineford, but at least he now has quite a more realistic outlook on life, simply not storming into fights with opponents he obviously cannot defeat - and not going into fights for no reason.
He gets that from his Grandfather former Vice Admiral Monkey D. Garp who also fits the qualifications to a horrifying degree such as throwing his Grandchildren into a chasm, tied to balloons, falls asleep sudddenly, and he once traded a large amount of gold for doughnuts.
Flint from Flint the Time Detective. Being a paleolithic child blessed with super strength, there wasn't a whole lot he needed to know or think about. Problem is he's time traveled to eras where the problems are more complex.
The trend is subverted with Taiki of Digimon Xros Wars, who has the strategic prowess to back up his title of "General" of Xros Heart, and looks like Sun Tzu next to the other protagonists. His successor Tagiru, on the other hand, actually plays it straighter than any of his Idiot Hero predecessors.
Teppei from B't X is very much this trope for a good chunk of the series. He eventually grows out if it, but it's pretty hard to root for him sometimes because of his pigheadedness.
Louie of Rune Soldier Louie — the despair of his adventuring buddies. The priestess who was directly ordered by her God to follow Louie as her personal Hero constantly cried "This is not my will!" and has no faith in his heroism until the final story arc.
Kamina from Tengen Toppa Gurren Lagann embodies idiot heroism to a degree rarely seen, perfectly happy to run off a cliff brandishing a sword in an attempt to fight an enemy in a giant robot, then dropping the fight to hit on the hot girl who just appeared out of nowhere. A good example is when, believing that sheer bloody minded determination can make up for any deficiency in skills or knowledge, he surpassed the identification system of his hijacked mecha with fighting spirit alone and proceeded to absolutely trash it through careless piloting. Another character-defining moment is when he stabs his Humongous Mecha with Simon's smaller drill-equipped mecha in an effort to combine them. If his universe didn't run on Rule of Cool, that would have been rather awkward, especially since the thing almost impaled his head. He has moments of intelligence - namely, trusting Simon to sink the Dai-Gunzan - but as a general rule, the only thing that keeps him from being killed with his idiocy is the fact that the universe agrees with his methods. Even that doesn't last forever.
The only thing that saves him is his smartness - he needs to act in such a way because it inspires everyone. He needs bravado, because everybody else has lived underground for their whole lives and nothing will change if he doesn't act in this way. He knows that Simon's mecha is special, and considering Simon went on to save humanity, then the universe, it was actually a pretty good bet.
It also helps that the mecha in question is powered by Hot-Bloodeddetermination and is capable of rewriting the laws of physics in the presence of enough of it. Kamina, and the rest of the Dai-Gurren Brigade, wouldn't be nearly as effective if the universe wasn't as crazy as they were.
Pretty Cure All Stars DX 3 showed what happened when you put all the pink heroines together in one place - because they're all leaders, trying to follow orders from another leader just throws them into major confusion.
Excel from Excel Saga is both a female example and a parody. A parody in that, besides being the henchgirl of an Evil Overlord trying to take over the city and not that nice of a person, she is this trope.
Iwata is another parody. He works for the other side, so he thinks he's a hero, but he's really destructive and nothing he does saves the city.
Relatedly, Poemi from Puni Puni Poemi is another parody, with the other characters and Poemi herself frequently pointing out her extreme stupidity.
Goku from Dragon Ball, the Trope Codifier for modern Shonen heroes. At the same time, he isn't a completely straight example of this trope since Goku is more naive and too trusting than outright dumb and does show a surprising amount of wisdom outside of fighting. It is hinted in Dragon Ball Minus that Goku's good heart came from his mother and not just from his head injury like most of his friends believe. There is also no real hint of Goku having brain damage given that he was raised in extreme isolation for most of his childhood which is why he is naive about the rest of the world and never really integrated into human society.
Yusuke being what appears to be an Expy of Goku with a JerkassDelinquent layer over him, also falls under this trope. He is severely, SEVERELY Book Dumb; he got a bloody 12 on a science test. Kuwabara is worse; he got a 7. However, Yusuke has far more common sense than the average shonen hero. Justified in Yusuke's case because he hardly ever shows up for school. You can't expect him to know material he wasn't in school to learn. Kuwabara is probably a straighter example of this, as even when he did study, he got a 53. And it's implied that at least part of the reason for his never doing so stems from the fact his mother, Atsuko, is frankly a terribly negligent parent. She spends more time out drinking, sleeping off hangovers, or cursing him out in a drunken tirade than doing anything responsible, to the extent that Yusuke himself has very little respect for her. To be fair to her, she was a party girl who got pregnant with Yusuke at the age of 14 and was evidently abandoned by both the father and her family.
Kuwabara's case could easily be considered justified as well, most of his school life was spent in a vain effort to beat Yusuke, who he considers his rival. Even during the times when he wasn't getting beat up by Yusuke, sleeping through classes or ignoring the teacher with paper airplanes and the like seem to fit rather well with his early personality. He subverts it by the end of the series by studying hard and getting into a good college.
Downplayed and a bit justified in in Naruto whose eponymous main character is primarily Book Dumb. Despite the fact that he is recognized at the bottom of his class, Naruto was capable of sneaking into the home of his village's leader, a wise and skilled ninja, and stealing a forbidden scroll, and learning a forbidden technique written on said scroll. He is capable, or "learns through his body" as Kakashi puts it. On the other hand, he is practically incapable of learning the basic techniques early on that all of his classmates learn. At one point, it is inferred that the Tailed Beast inside of him is disrupting his anatomy, resulting in at least a few of his academic failings. Regardless, as to his personality, Naruto is still an overly spirited and loud-mouthed individual throughout the first half of the series.
In Shippuden, he gets a bit better at making strategies, to the point where his own sensei was surprised at getting caught off-guard by him. Naruto's newfound tactics cause most of his acquaintances and allies to marvel at how he's grown. Sakura, however, berates him from holding onto a perverted technique and believes that he still hasn't really matured. Due to this particularly unhappy childhood, Naruto is also rather good at relating or identifying with people who grew up in similarly negative ways, so something in his brain is definitely churning. He's also darn good at Shut Up, Hannibal! speeches.
Tamaki Suoh from Ouran High School Host Club is so dumb, he concludes that his possessiveness towards, his nervousness around and wanting to kiss the first real female friend he's ever had is because he wants to be her... get ready for it... father. His True Companions know he is dumb, but assumed this was just a metaphor. Then they learned that no, he really thinks that is logical. Even though Tamaki's hobby and job is entertaining women, so he knows what romance is. He's the school King of romance.
Luckily, he isn't really the King of the Host Club. We all know who the Shadow King is.
Also his idiocity only relates to relationships, which is probably explained away by his own family history. He's actually second in his class, right after Kyoya.
Gourry Gabriev from Slayers is a personification of this trope and is (as is all of Slayers) at least partly parody. He doesn't know basic facts about... well, anything other than food and fighting, and is prone to forgetting minor details like the names of his best friends, and that time a few months back when they saved the universe in an epic battle of good and evil. But his combat skills are such that he can assess the skills of other characters accurately just by glancing at them and can hold his own in a fight against everything up to deities when he has his legendary Sword of Light. Word of God states that in addition to his skill with the sword, he's got the innate talent to be a truly frightening sorcerer — but he's too dumb to remember the necessary magic words.
Gourry subverted the trope, in a way, in the one time throughout the entire series that he showed any signs of intellect; when Xellos is finally revealed to be a Mazoku, good old Gourry is shocked to hear that no one else had figured it out. He had known since the beginning.
Speaking of Slayers, this is somewhat subverted in one episode ("Geoffry's Knighthood") with Geoffry, whose mother employes Lina and Naga to accompany him on his quests. Although Geoffry is an "idiot", he fails somewhat on the "hero" — all of the badass fighting is actually done by his mother ("disguised" by a bag over her head). However, the final Boss is genuinely terrified when Geoffry appears...
That's got little to do with Geoffry, though. As one comes to expect of Slayers, there's a new layer of parody here. The main boss is Geoffry's Dad!! And the reason for the fear is eminently logical; the man KNOWS his maniac of a wife will be by to defend Geoffry. That reaction is pure channeling of Harry Mudd, my friends.
Gourry also subverts this trope in the original novel series where he is far more capable and his ditziness is closer to Obfuscating Stupidity. It's the anime series that flanderized this idiot trait to max.
Invoked Trope in the Mahou Sensei Negima! manga. The hero Negi Springfield is intelligent... intelligent enough to worry and hesitate. The Rival has stated that Negi needs more "idiocy" to be a better fighter (Negi goes on to ask his trainer how to be an idiot, like him).
Compounding on this, according to flashbacks, Negi's legendary father Nagi appears to be a textbook example of an Idiot Hero. (For example: instead of memorizing his combat spells, he wades into battle with crib notes in one hand.) However, he turns out to be somewhat of a deconstruction. Yes, he stopped the Lifemaker and CE twenty years ago... but he never even seemed to realize what their motivation was and had no plan to solve the problem they were attempting to address. Instead, the next generation is the one that has to solve the actual problem. Just look back at Nagi's final beatdown of the Lifemaker after having read chapter 325.
Kotaro definitely counts; he even prides himself on it.
There's also the Baka Rangers (with the exception of Yue), who have the lowest grades in the class but make up for it with their physical skills. Especially Asuna, who is the worse scoring student and the deuteragonist.
Goku from Saiyuki. The most recent manga chapters suggest that he is slowly maturing, however.
Jil, from The Tower of Druaga. In battle, he's been known to do exactly the opposite of what his more experienced party mates consider the most sensible tactics. He's also one of the only characters on the quest for the purpose of slaying Druaga, and has been called a "Justice Freak". Comes complete with an Aloof Big Brother.
Arika Yumemiya from Mai-Otome, although in her case it's more a matter of being inexperienced with the world. She gains some confidence and competence as the series goes on, but while almost everyone else in Mai-Otome Zwei has become Older and Wiser, Arika lapses back into Idiot Hero mode.
She's even dumber in the manga, although part of the reason why she isn't good at anything except Butou is that, not having Sergay's sponsorship, she has to work several part-time jobs to support herself.
One of the older examples of an idiot hero is Kouji Kabuto from Mazinger Z. Excellent pilot, marvelous sniper, big ass Determinator, pretty good with motorcycles — but his little brother, his girlfriend and his Gonk rival keep correcting and bitchslapping him (verbally and literally) for his lack of tact, sexism and big mouth. Nevertheless, Kouji, unlike many similar hot-heads, may be exceptionally bright if he allows himself think things through. He is more of a Genius Ditz.
His successor and adoptive brother Tetsuya from Great Mazinger is another determinatorBadass and he is slightly more mature and less sexist than Kouji, but some of the stunts he pulls due to his pride, hardheadness, jealousy, obsessivity and unability to listen to good advice are mind-baffling. And then you have he can be a tactless jerk when he gets angry or upset, and it is not wonder how often he has been on the end of several Jun's Armor Piercing Slaps.
On the other hand, the remainder male main character of the Mazinger trilogy -Duke Fleed from UFO Robo Grendizer- averts the trope. He can be somewhat rash sometimes, but usually he thinks before of acting.
Sonsaku Hakufu from Ikki Tousen is a total idiot. She knows it herself, even though she hates being called stupid, which happens so often that she should actually be used to it. Her inner dragon makes her a very strong fighter, though, and like all other female warriors in the series she is hot, which is a combination that definitely helps her to get more respect from her peers.
Yoichi, the lead of Samurai Harem: Asu no Yoichi. Since he spent most of his life training in the mountains, all he thinks about is fighting.
While the titular character of The Law of Ueki does have several brilliant insights, especially in battle, for the most part he seems rather simple-minded and clueless. On first meeting Tenko's gigantic dog-like form, he thinks it's just a really big chicken.
Maka and Soul embody aspects of this character. While Maka isn't stupid, she can be quite naive yet wholesome. Medusa takes advantage of this. Soul was more this during the beginning of the manga, recklessly disobeying Maka to devour souls. Both evolve however.
Arguably Kid falls somewhere between the two (which he would not like) - he stops to think at times, but does make downright stupid decisions, at least in part thanks to his very single-minded way of thinking.
Taken a step further in the 2010 manga retelling of the series. To give an example, in one episode he uses an Air-Vent Passageway to ambush a fellow Gundam Fighter's boxing opponent and take his place. In the manga version, he ends up in the opponent's room by accident, attempts (pitifully) to talk his way out of it, then beats him up in self-defense and decides to take his place on the spot.
Michiko from Michiko & Hatchin arguably qualifies. She tends to make quite questionable decisions while she runs away from the police and tries to find her old boyfriend — her flashy appearance not even being the stupidest. Fortunately, the girl she took with her, Hana, is actually quite a bit more intelligent. Unfortunately, Michiko never listens to her.
Alice L. Malvin of Pumpkin Scissors insists on going after evil no matter the odds, so obviously, her subordinates have to stop her from just charging head on into battle and come up with a more effective, less risky plan.
Character Development into All-Loving Hero? Check. His development didn't stop when he put on the armor and he spent part of s3 after that as The Atoner and then saved all of his friends and went on to save the world from Darkness, after he helped Alexis, Chazz, and Syrus find the paths they wanted to follow in life. And he's hardly a loner when he lives with Yubel in his soul, his entire deck of spirits in his pocket, and Pharaoh and Banner in his backpack.
Crow from Yu-Gi-Oh! 5D's, though it's only to assist his friends. Wants to prove to them that he doesn't want to be a 3rd wheel or a burden to them.
Yuma from Yu-Gi-Oh! ZEXAL is season 1 Jaden all over again. And despite being the protagonist, he has the dueling skills of an amateur, and Astral has to guide him through a lot of his duels.
In his defense, Yuma slowly becomes a better duelist over the course of the show.
However his Idiot Hero tendencies rear their ugly heads when he doesn't tell anyone that Shingetsu is a Barian (Shingetsu asked Yuma to keep it a secret), which comes to bite him in the ass when Shingetsu turns out to be Vector and the fact that Yuma kept Shingetsu's true identity from Astral ruins Yuma's relationship with him.
Yuya from Yu-Gi-Oh! ARC-V is another subversion. While he initially seems like one, he comes across as having an average intelligence than actually dumb. He mentions being bad at Math and Science but apparently does decently in other subjects. His main dumb moments were having trouble with Pendulum Summoning at first, though using card game is treated as being a lot more difficult in the show anyway.
Natsu from Fairy Tail demonstrates many elements of this. In a semi-subversion, despite being a Hot-BloodedDeterminator, prone to causing extreme collateral damage and social obliviousness (the reader is lead to believe that its because he was raised by a dragon), he's perfectly capable of fighting intelligently and wins through unorthodox use of his abilities at least as often as through sheer power (of course, most Shonen protagonists are like this).
Sting of Sabertooth turns into one after the Grand Magic Games.
Gon from Hunter × Hunter. The only reason he probably hasn't died yet is because he comes up with creative ways to stay alive during a fight. He forgets things often, is easily fooled, rushes into things without considering the consequences, often even disregarding the consequences just to get what he wants.
Baccano!'s Firo Prochainezo is one, and also thoroughly aware of it.
Natsuru, main character of Kämpfer, in the anime. S/he was already dumb as a brick with special needs in the manga and light novel, but the anime went Up to Eleven and turned hir into a character thinking exclusively with hir non-existent penis... while being Oblivious to Love. Consequentially, s/he spends most of the series just sorta standing there with a dazed and confused expression, not even bothering to do much fighting or noting the Obviously Evil characters, or wondering what the smeg was going on with the secret magical girl war s/he was drafted into, or why the hot girls kept blushing around hir and taking hir out to eat... S/he is also a Pinball Protagonist, doing absolutely nothing of relevance whatsoever other than obliviously make all the lesbians in the room damp.
Every hero in works by Mitsuru Adachi (most famous works include Touch, H2, Miyuki, Cross Game) would be labeled under the "Lovable Idiot" category. Though they often act like Idiot Jerks they're actually quite smart and tend to be sarcastic and smart mouthed.
America from Axis Powers Hetalia is the poster boy—er, nation—for this trope. He's a Big Eater who is loud (occasionally considered annoying by the other characters and/or fans) and comes up with plans like "building a giant superhero to stop global warming." He is repeatedly told to "read the atmosphere," upon which he looks for it in a bookstore. A quote from the dub:
America: Someone told me that my elevator doesn't go all the way to the top floor, but I don't even have an elevator!
One of his favourite phrases is "Mochiron, hiiro sa!" which translates to "Obviously, I'm the hero!" It's even in his character song, "~WDC~ World Dancing".
Shirou, the protagonist of Fate/stay night has been known to cause large red spots to appear on the foreheads of viewers of the anime from facepalming so much. In a war involving ancient epic heroes and massive magical powers, Shirou is absolutely and completely determined to not let his fighting companion get hurt (even though she is infinitely more competent at fighting other servants than he is).
Touta Matsuda from Death Note is a non-protagonist example. He is the most inexperienced member of the SPK, due to lacking in seriousness.
Arata from Arata Kangatari fulfills this trope to a T, having many of the stereotypical traits expected of an Idiot Hero...except for the fact that he turns out to be the Decoy Protagonist early on and is relegated to the main character of the B-story.
While none of the characters from JoJo's Bizarre Adventure are explicit idiots (in fact, several can be considered border-line geniuses considering what they do with their powers) there is one shining example: Okuyasu Nijimura from Part 4, who is so stupid Josuke defeats him literally by ducking (and then has to save his ass when he follows Josuke into a house specifically stated to have a killer stand in it just to give him respect). In fact, both Josuke and his own brother state that if Okuyasu wasn't such an idiot, he would be a very dangerous threat (what with the whole "Stand-erases-matter" thing).
Rurouni Kenshin: Sanosuke Sagara is a reckless street fighter who mostly either concerns himself with mooching off his friends or finding another fight, but often shows enough insight and reliability to be Kenshin's partner.
Touma Avenir from Magical Record Lyrical Nanoha Force is another naive example. Sure, he's not an idiot at all. But his naivete makes both Section 6 and Eclipse infectees so irritated. Especially the latter. Considering that his mentor and idol is Subaru Nakajima, herself not the brightest tool in the shed, it's obvious where Thoma's naiveté is coming from (Subaru, at least, has Teana to set her straight).
Teru of Castle Town Dandelion has this in spades. His heart is ultimately in the right place and has the appropriateRoyalty Superpower to boot, but he suffers from a severe lack of common sense and what appears to be chuunibyou (justified by being an elementary schooler). In his introductory chapter, his kindergartner sister Shiori is more competent than he is, remembering to bring the bag, getting a dog to stop barking at them, and reminding Teru that he had left the shopping list in his pocket. The one point where he was actually useful was at the end of the errand, where he fends off a stray dog who wanted to attack Shiori for the groceries.
Ryuu Yamada from Yamada-kun and the Seven Witches is actually a slightly Harry Potter-like example in that his overall intelligence clearly isn't that low, but he is rather Book Dumb and unmotivated and tends to rush into fickle situations without thinking about the consequences or considering making a game plan.
Set up and then averted by Issei in High School D×D. Given his behavior in the first few episodes/chapters, you could be forgiven for thinking he was shaping up to be a textbook example. However, he quickly proves to be a fast learner, and completely aware of how far over his head the Devil world is, leading him to defer to Rias long after he becomes able to look after himself. He never quite shakes the nickname "Power Idiot", but that's got more to do with his fighting style than actual intelligence.