Harry/Cho was swiftly sunken in this book, along with nearly all of its supporters.
Many people realized that the scene with Harry and Ginny eating chocolate in the library (in the book) was a torpedo heading directly for the Harry/Hermione ship.
Accidental Innuendo: Draco makes a barb saying "we'll be dogging your footsteps" (referencing Sirius turning into a dog). Dogging is a UK slang term for having sex in public places, knowing there are people nearby.
Luna, in the mistletoe scene in the Room of Requirement: Legitimately worried about Nargles, or covering for her dashed hopes that she could finally snag a kiss from someone?
Because of how little characterization she had, Marietta Edgecomb is subjected to this. Was Cho telling the truth that she only ratted out the DA because she didn't want her mother to lose her job, or was Cho lying in order to cover her ass? Or, like Cho in the movie, was she finally forced to crack under the effects of Veritaserum?
When Sirius encouraged Harry to set up the DA, what was his main reason for doing so? Was it to make sure Harry and his friends could protect themselves against Voldemort, or was it the chance to rebel against The Man once again, through his godson? Or maybe a little bit of both?
Molly asked Moody to find a Boggart in Grimmauld Place with his magical eye, and he found it. Did Moody see the Boggart in its (if such existed) true form? Or did Moody see it shapeshifting into a form of his own greatest fear? Considering Boggart only shapeshifts when it sees its victim, it could very well be the former, which means Moody might be the only known character ever in the franchise to see Boggart in its true form! Too bad it's never mentioned again.
Angst Dissonance: Harry's angst after Cedric's death and being left out of the fight against Voldemort by the Order (because the kid who saved their asses four times already can't possibly be useful), while justified, was still an annoyance to many readers.
It's likely that Harry has PTSD in this book. He watched a classmate die, was tortured, and forced to fight an older wizard with over 40 years and a lot of power on him. His angst is perfectly reasonable, but according to some doesn't make it any less obnoxious.
Subverted in the film, a rare instance where some prefer the film's portrayal to the book's. Harry's angst is downplayed to relative hesitation and uncertainty of himself as a leader. One little pep talk later and he gets over it.
An act like that should be considered Dark Magic and vigilante justice, with some even comparing it the Bat Brand.
Was Umbridge raped by the centaurs? If she was, did she deserve it?
And the matter that wracks many an entry in the Headscratchers page: was the Prophecy worth all this convoluted cloak-and-dagger stuff that forms much of Harry's angst throughout the book and ultimately got Sirius killed?
Designated Villain: While Sirius' treatment of Kreacher is painted in a poor light, it should be noted that while he was mostly just grumpy and uncaring towards him and never abused his authority or fired the house elf, Kreacher was the antagonist in their relaionship, constantly and ubceasingly insulting him and his friends, including calling Hermione a mudblood, and plots his murder at the first opportunity. It's little wonder Sirius doesn't treat him with love and respect.
Snape has a very sympathetic backstory, but Fandom sometimes will go out of its way to blame every one else and sugarcoat his part things that went wrong in his life.
Marietta as well to a smaller extent (since we know very little about her). The fandom often likes to imagine her as a tortured girl afraid of her mother losing her job at the Ministry - when there's very little evidence pointing to that in the book. Whatever her motivations, she still went to Umbridge of her own free will knowing people would be punished.
Ensemble Darkhorse: Luna Lovegood. Originally only to have importance in this book, by the end of the series she is the sixth member of the True Companions group and the only non-Gryffindor to be in it. Luna's so popular, she managed to make her Quibbler critters a popular source of Arbitrary Skepticism jokes in the fandom, and many fans were disappointed that she never discovers a Crumple-Horned Snorkack during her time as a Magizoologist.
Remember Molly's Boggart? Fred dies during the Battle of Hogwarts.
Harry tells Sirius that Umbridge is no different than a Death Eater with some of her tactics. Guess who she ends up siding with two books later?
Fred and George shoving the unpleasant Slytherin Montague into a Vanishing Cabinet is funny at the time, but less so after Half-Blood Prince, where we learn that this incident alerted Malfoy to the Cabinet's abilities, so he spent the year fixing it in order to let Death Eaters into Hogwarts, including Fenrir Greyback, who mauls Fred' and George's brother Bill.
After the death of Sirius, Dumbledore tells Harry, "I know how you're feeling". In Deathly Hallows, we find out that he has also suffered the loss of family members largely through his own fault.
Zacharias Smith's "I don't think Expelliarmus is exactly going to help us against You-Know-Who, do you?", given that this spell is what finally brings Voldemort down.
Harry says sarcastically to Malfoy, Crabbe and Goyle: "I'm terrified now. I suppose Lord Voldemort's just a warm-up act compared to you three?" In Deathly Hallows, those three are a warm-up act before the final confrontation with Voldy.
Regarding the Ordinary Wizardy Levels tests, there's one concerning the grades. The lowest grade possible is "Troll", presumably meaning that the student has the aptitude level of a regular in-universe troll. Time marched on and the concept of the Internet troll arose. This gives the grade a wholly different but still perfectly applicable explanation in that a student reaching that grade has got to be doing it on purpose to get a laugh at the examiners' expense, i.e. trolling them.
Cho for calling Hermione out on disfiguring her only friend and pointing out that she should have told the members of the DA that the sign-up sheet was jinxed.
Sirius is submitted to a lot of this in Harry Potter and the Order of the Phoenix with many characters mocking him for his "uselessness" to the Order while Hermione and Mrs. Weasley note how reckless he is. From what we see, Sirius was capable enough to be two years on the lam, disciplined enough to keep his sanity while imprisoned in Azkaban and his recklessness stems mostly from being cooped up in the house he hated and being submitted to Condescending Compassion from everyone around him, with Hermione expressing doubts about the advice of a full-fledged wizard. Indeed, had Harry listened to Sirius instead of Dumbledore or others, by using that Mirror he could have verified Sirius being safe and alive instead of charging at the Ministry. Dumbledore outright admits that Sirius was right for wanting to involve Harry on information with the Order.
Launcher of a Thousand Ships: Since this introduced some new characters, it also launched a few new ships. Most prominent was Harry/Luna and Neville/Luna.
Love to Hate: Few, if any, fictional characters are as supremely and utterly despicable as Dolores Umbridge.
Memetic Mutation: Umbridge's "I will have order!" She says it once in the movie and never in the book. Tellingly, by the time the next film was made, they saw fit to include an Umbridge doll in Weasleys' Wizard Wheezes which constantly squeals "I will have order!"
Memetic Psychopath: Between the jinxing of Marietta and the popular theory that Umbridge was gang-raped by the centaurs with her having a victorious smile afterword, some fans like to think there's a sociopathic side to Hermione.
Dolores Jane Umbridge has long since passed over it. She starts by cheerfully calling Harry a liar, forcing him to cut into his own flesh and do lines with his own blood, tries to sabotage Gryffindor's chances of winning the Quidditch Cup by removing three of their best players, cruelly mocks Hagrid's teaching, gleefully tries to sack Professor Trelawney (and thoroughly enjoys her despondency), takes over from Dumbledore, tries to arrest Hagrid for no reason, nearly kills McGonagall in a sneak attack, and finally tries to used the Cruciatus Curse on Harry. Oh, and it turns out she was the one who set Dementors on Harry and Dudley in the beginning. So, take your pick. Oh, and guess what? She gets worse in Deathly Hallows.
In the movie, there's a scene that shows that she forces first years to use the Blood Quill.
UncleVernonDursley crosses it in the novel (but not in the film) when he tries to kick Harry out of the house, after finding that he would likely die in the hands of Lord Voldemort if he did this.
In his very slight defence, his reasoning for doing so are for the sake of his wife and son and their safety, not specifically because he wants to see Harry killed (though he wouldn't likely care if his nephew was killed). Vernon, not understanding magic and that he and Harry are both better off with Harry there, thinks he'd be protecting his family by getting rid of Harry.
What can't be defended, however, is that he very clearly wishes death upon Harry, as he all but outright stated earlier in that scene he hoped the Ministry of Magic executed him.
Some found Harry's meltdown in Dumbledore's office after Sirius died to be this. It looked less like he was venting his anger and more like one of Dudley's temper tantrums in the first book. The earlier rant in Grimmauld Place is like this too, with most of Harry's dialogue in block capitals - and some fans have jokingly nicknamed it 'Caps Lock Harry'.
Umbridge is so blatant and sometimes cartoonish with how evil she is, it's any wonder it took so long for her to meet her downfall.
The film version of the duel between Dumbledore and Voldemort; visually stunning? Absolutely. Intense? Oh, yes indeed. But there is that one part where Voldemort looks like he's belching a giant fireball at Dumbledore...
The release of the fifth book and the "Snape's Worst Memory" chapter has become the defining image of James Potter's character among the book's fans and especially Snape's fans, rather than the positive way he was regarded and described in the first four books as a great wizard, loyal friend and good father. The fact that Rowling never bothered to flesh out James's life with more flashbacks showing him in a more positive light (after he "grew out of" his bully phase) doesn't help.
Hermione jinxing the DA sign-up sheet so that it permanently disfigures potential traitors has made some readers see Hermione as a sociopath with a Judge, Jury, and Executioner mentality.
The Death Eater whose head — just his head — was regressed to infancy.
When those brain things attack Ron in the Department of Mysteries, shortly after Ron was struck by some curse that left him blabbering like a tiny child.
Umbridge. She sits in as a voting member of people's trials? She's allowed around children?! Her tactics are so draconian she may as well be a Death Eater. And Harry says so himself: "She's foul enough to be one."
The bureaucracy that is the Ministry Of Magic for turning a blind eye to the atrocities Umbridge committed. The Death Eaters had taken over the Ministry when Umbridge was at her very worst but this in no way excuses her actions.
The Ministry's claim that they need to interfere in Hogwarts in order to fix various problems with the way the school is run. They're doing it more for personal gain and paranoia about Dumbledore but some argue that if a real school was run like Hogwarts (for example, letting a teacher bring dangerous animals to class), it probably would have been shut down.
After more than a book length's buildup of the Department of Mysteries, which was foreshadowed in Book 4 and mentioned suggestively like many Chekhov's Gun in Harry Potter's books, we get a spooky chapter describing the various weird rooms and the possible secrets they may contain. Then it ends up being a simple background for a fight scene and is never mentioned again in the later books. First time readers expected complex World Building and more details about the magical lore from the place but it becomes entirely marginal despite the buildup.
Despite being the titular group, very little of the book focuses on the Order or its backstory beyond Noodle Incident. The novel mostly focuses on Harry's Emo Teen phase and Sirius' backstory as the White Sheep of a Pureblood family. Many felt disappointed that the book, and later books, don't tell us more about how effective they were, what they actually do to help people and most importantly when and how the Potters and Longbottoms thrice defied Voldemort.
Unintentionally Sympathetic: Everyone plays Marietta's betrayal of the DA as a Moral Event Horizon, but she did it out of fear that her mother would lose her job. Of course, her mother's job is being a stooge for Umbridge in installing surveillance over the Floo Network which more or less leads Harry and others in the fandom to string the World's Smallest Violin (especially since Umbridge has made it harder for genuinely struggling and upstanding folks like Lupin to get legal work).