Harry's kind of a jerk.
Actually, that's a bit harsh. He's not a jerk any more than the next teenager, although that's not saying much. But Harry Potter's not particularly interesting, either. He's a poor hero, either in ideological or literary terms. There are lots of reasons that I think a person could dislike J.K. Rowling's books--they're formulaic, trite, cloaked in nostalgia, and not terribly original. If you've got to read youth fantasy, you can do much better...
...which is your cue to note that I am (like everyone else who dares to dislike Rowling's work) humorless, elitist, and needlessly contrarian. If I'm not cloaked in child-like wonder at the series, the conventional wisdom goes, there must be something wrong with me. And at this point, fine. If you can't understand why someone else finds the books tedious, I'm not going to argue with you. Come back later, and we'll talk about something less distasteful for both of us.
Anyway, as I was saying, I think there are lots of reasons that someone could dislike the Harry Potter books, but for me it's always come down to Harry himself. Because he's kind of an unpleasant little sod, isn't he? It's easy to miss, because he's portrayed so sympathetically (although I would never call Rowling a cunning wordsmith, her prose is far from wooden). But I always find, reading the books (and I have read them all, some of them twice), that it's hard to rectify the oddities of Harry's actual character with the character that Rowling thinks she's writing.
Let's recap. When we first meet Mr. Potter, we're told that he is famous because Voldemort's attempt to killed him backfired. Over the next few books, Harry takes place in a number of conveniently year-long escapades, mostly because the recovering Voldemort insists on tying up this loose end, and not because of any action on Harry's part. Eventually, Rowling strengthens this plot from simple revenge by adding a few macguffins: a prophecy that links the two, as well as some magical handwaving like psychic links. But what increasingly becomes clear is that Harry himself did not actually defeat the Dark Lord--I believe one book credits his parents' love (a mushy, underwhelming plot device if ever there was one) with deflecting the death curse.
But he's got other virtues, right? Not really, other than a garden-variety bravery that comes standard on every young fantasy hero. Harry's a poor student, and if he doesn't cheat than he's at least willing to come very close. His selective ethics are discomforting. He has a poor temper, and regularly fights with the people around him over very stupid things. His magical skills don't seem to be noticeably stronger or more refined than the other students. Hermione regularly outstrips him in every category, except for Being An Insufferable Stereotype of a Teacher's Pet. Harry is, in other words, a flawed (and frankly, unattractive) character.
Now, I personally welcome the average hero. I'd love to read a book about someone in a fantastical universe who is not either The Hero of Legend, an Unlikely Savior, or A Being of Vision. But Rowling is not writing those books. Instead, she's writing stories of a young man with a destiny, and her other characters insist on responding to Harry as if he's someone great. People are impressed with Harry, even though by his actions and attitudes he is really quite unimpressive. In general, he overcomes obstacles through either the help of others, or a magical deus ex machina which owes nothing to his own skills or abilities. It's almost sad, really, that in a world filled with wizards and monsters, someone like Harry Potter can even flag down a taxi.
I would like to believe, honestly, that this is the joke. I'd like to think that the Harry Potter books are really a satire about some poor kid in the wrong place at the wrong time, stuck trying to live up to the sadistic prophecies that always anchor sub-standard fantasy novels. His fate as an orphan is just one of the cruel, ultimately shallow literary devices that a fourth wall-breaking protagonist could bitterly lament. But I see no evidence that the author is not playing it straight, and the discontinuity between Harry and the events around him is glaring. Why is he a hero? Only because the book says he is.
At the core of a great deal of (bad) fantasy is this concept of "the chosen one," and Harry Potter slots right into it. If anyone is going to sling around accusations of elitism, it should probably start there. Yet since most readers of Rowling's series are probably blissfully unaware of either the overworn tropes of genre fiction, or the progress that's been made in overcoming them, this context is lost. It's more depressing than anything else. These are not bad books, when all is said and done. They're technically well-written, smoothly plotted, and deftly marketed. But let's face it: they're pulp, and not even the best pulp out there. Considering that Harry's fans are unlikely to ever read another fantasy novel--that they'll never graduate from Hogwarts School of Magic to Unseen University or the college of New Crobuzon--they deserve a better class of protagonist.