Specific to the book (or applying to both film and book):
Alternative Character Interpretation: Deathly Hallows caused many people to see the characters in a different light, including but not limited to Dumbledore, the Malfoys, Ginny, Snape, and Voldemort. It might actually be easier to list who isn't seen differently because of this. See here.
The Deathly Hallows and the entire new series of mysterious artifacts feels like this to readers who felt that Book 6 had established the Horcruxes as the main MacGuffin to hunt down and were amazed that Rowling introduced a new mystery on top of it. Some felt that if the author had introduced this in Book 6 or at least Foreshadowed it, it would have worked better. It's true that Harry's invisibility cloak never tears or falls into disarray was mentioned before, and it does fall into the pattern, much like the rest of the future plot devices in the books. It's also true that Harry is quite reasonable in taking the power of his Cloak for granted, since he has no other frame of reference. But various other characters do have a frame of reference — Mad-Eye Moody, who owns two, springs to mind — and none of them ever comment on its status or unique qualities.
The concept of Wand Lore. It's true that in the first book "The Wand Chooses the Wizard" indeed, but the idea of the wand changing allegiances during a wizard duel was never suggested once even if Wizard dueling was mentioned several times and discussed many times in earlier books and it mostly feels like a cheap and obvious way for Harry to defeat Voldemort without killing him, by using his signature "Expelliarmus."
The fact that Fiendfyre could destroy Horcruxes was not mentioned until after it burst out and attacked everybody. Hermione handwaves it by saying it was too difficult and dangerous to attempt (and near the beginning of the book she did say that many of the substances that destroy Horcruxes are impractical), since its flames hunt down its victims. The fact that the object is a Horcrux that is mentioned as being nearly indestructible several times and which Dumbledore got cursed trying to destroy, to quote Ron Weasley, "really gives a feeling for the scope and tragedy of the thing, doesn't it?" The film seems like it's trying to remedy this by having Harry stab the diadem with the Basilisk fang before throwing it into the fire. Just to be safe.
Ron enters the Chamber of Secrets. By learning to speak Parseltongue. Only not really because he was only imitating the sound of it. And it let him in anyway. This isn't mentioned once before in the books and even Dumbledore had difficulty understanding snake speech, but somehow Roonil Wazlib has a facility for language.
The Deluminator comes off a bit like this, as its previous appearances gave no indication that it could be used the way it is in this one, but it's less so compared to the other objects in the books.
Die for Our Ship: After announcing that Harry and Hermione would not become a couple, J.K. Rowling was actually getting threats of boycotts (as well as death threats) from rabid Harry/Hermione shippers. She apparently brushed them off with this book as a big Take That!.
Draco in Leather Pants: Well, it had to happen eventually. After the defeat of Voldemort, Draco reforms and, in the epilogue, ends up being just another parent sending his wizard offspring to Hogwarts.
Ending Aversion: Downplayed. The epilogue wasn't received well, what with Harry naming one of his children AlbusSeverus and how half-hearted it came off as compared to the rest of the book, but the final few chapters of the story itself were well-received.
Fan Fic Fuel: Thanks to the Trapped by Mountain Lions nature of the plot, parts of the plot such as the full extent of Voldemort's takeover of the Ministry, the Order of the Phoenix forced to go underground, and Neville reviving Dumbledore's Army to rebel against the Death Eater rule at Hogwarts became this.
Some Harry/Hermione shippers think that JKR wrote the "nineteen years later" epilogue primarily or solely to spite them by making "Harry and Hermione get together after Hogwarts" stories uncanonical even though she wrote it years before the first book was released (authors frequently plan their endings or have them written ahead of time). A number of fans reject the canonicity of the epilogue to get around this, leading to the phrase "Epilogue? What Epilogue?".
Molly killing Bellatrix, an act which many believed (and still believe) belonged rightfully to Neville.
More than a few fans also explicitly reject the idea of Harry naming his youngest son "Albus Severus Potter" after Dumbledore and Snape, considering the sheer amount of emotional anguish that both characters put him through. Most of them prefer to believe that he would have given that honor to people that actually comforted through his turbulent teenage years. It's not uncommon to see fans suggesting "Rubeus Remus Potter" as an alternate name.
"Funny Aneurysm" Moment: During the wedding, Fred comments that when he gets married, everything will be mellow and he'll put Molly in a Body-Bind Curse until it's all over. While this sort of speech is typical of Fred and George, it becomes quite tragic after you've finished the book.
Genius Bonus: Hermione was named after the character from Shakespeare's The Winter's Tale, which begins with a man who believes that his wife (named Hermione) is cheating on him with his best friend. It turns out Ron always feared Hermione liked Harry more. There's also the fact that Hermione is the female form of Hermes, being the god of cunning in many Hellenistic cults.
Remember Molly's boggart from Order of the Phoenix? It comes true with Fred's death at the end of the book. Even worse, you might remember that the boggart took the form of both Fred and George...meaning that even in her worst nightmares, Molly couldn't picture Fred and George being separated by death. Many fans wish J.K. Rowling couldn't have imagined such a scenario either.
Early on, Harry says that parents shouldn't leave their children unless they have to, after refusing to allow Remus to come with them on their search, because Tonks is pregnant with their child. Remus and Tonks are killed in the Battle of Hogwarts, leaving their son an orphan, meaning that Remus did have to leave his son.
The scene with dead, injured, and mourning students in the Great Hall can be harder to watch now in light of Sandy Hook and other major school shootings. Even worse, the book was released only two months after the Virginia Tech mass shooting.
Remember Neville pulling out Gryffindor's sword and beheading Nagini? Yeah, a Moment of Awesome, right? Well, the trailer for Fantastic Beasts: The Crimes of Grindelwald reveals that Nagini used to be a woman. Either Neville was committing what could be considered murder, or he was giving her a Mercy Kill. Granted, he didn't actually know that he was doing that, as no one knew that fact except for Voldemort, even the audience before that film came out almost a decade later, so he's innocent either way.
And then there's the bit with Bathilda Bagshot where Nagini is possessing her. She got to be human again just for a time.
Reading the scenes in Chapter 8 that involve Elphias Doge (who is always referred to as 'Doge') is much more hilarious if one is familiar with the "Doge"meme
Aunt Petunias actress Fiona Shaw would go on to play a witch on True Blood which can now be seen as Aunt Petunia getting what she wanted all those years ago.
Inferred Holocaust: The fate of many Muggle-borns at the hands of the Death Eater-controlled Ministry of Magic. J.K. Rowling revealed on Pottermore that many of them did in fact die while they were imprisoned, which was why Umbridge was given a life sentence.
Informed Wrongness: Hermione once again plays Sirius death at Kreacher's hands a cautionary tale and more or less states that Sirius suffered a Karmic Death for his poor treatment of house-elves and even equates how Sirius treated house-elves with how Voldemort treated him. The stated false-equivalency strikes many readers as off-base since Voldemort exploited Kreacher as a stooge and left him to die, while Sirius was merely cold and indifferent to him and would certainly not have killed him or treated him in such fashion, and the idea that being cold and indifferent to someone makes one the same as the main villain in the eyes of Kreacher makes it come across as completely unjustified.
Aunt Petunia. Yes, there's no excuse for the way she treated Harry, but in the book we see that her "you're a freak" attitude grew out of insane jealousy. Imagine being a little kid, seeing all the goings-on at Platform 9-3/4, and knowing you'll never, ever get to be part of that world.
Cadmus Peverell. He indeed committed hubris by defying death but, differently from his power-thirsty brother Antioch, he did it because he couldn't let go of his fiancée's untimely death, which is a much more sympathetic motivation. The fact that he was eventually Driven to Suicide by his wish makes him look like the victim of Disproportionate Retribution.
Snape who made terrible choices as a teenager growing in a poor home and abusive childhood and making his life so miserable that he spent the res of his life in self-loathing, never allowing the world to "see the best part" of himself forever pining after a fifteen year girl who never returned his love, married the person Snape detested, and had a child who is the living example of everything he ever missed must have been humiliating for him to deal with, to the extent that even near the end of his life, when he tears a photograph of the Potter family to keep Lily, he's unable to move on. The fact that Dumbledore strung him along to do a truly thankless task and that he died believing that Lily's sacrifice for Harry would have been in vain is pitiable.
Like You Would Really Do It: As the books progressed and got darker, Rowling was sometimes asked, "what's Harry going to do after Hogwarts?" She would toss in a "How do you know Harry is going to survive?" People commonly invoked this trope in response. Harry does die... just not permanently.
Neville Longbottom will reject your bribes and kill your pet. Neville Longbottom uses Basilisk venom as soy sauce. Neville Longbottom would have done it in four books. Neville... you get the idea. Chuck Norris's patronus is Neville Longbottom. Thereby making Neville the Wizarding World equivalent of Chuck Norris.
Neville killed over a dozen followers of Voldemort, saved his friends from a large snake that had Dark Magic in it, and charged at Voldemort himself with a magic sword that only the truly courageous can wield, all in one day. Which is all technically true.
Moral Event Horizon: Throughout the books, Malfoy's sidekicks Goyle and Crabbe were mostly ineffectual rivals to our trio, but towards the end, Crabbe actually goes evil and almost kills the three, attempting to hit them with the Killing Curse, even when Draco is dead set against it. After this, his self-inflicted death can't be seen as anything other than karma.
Ship-to-Ship Combat: Or rather Ship-to-Ship Last Stands, as this book effectively crushed the hopes of numerous shippers, but especially Harry/Hermione shippers. However, true to the nature of shipping, many prefer their ship than what was written in the books. Word of God (Rowling and others) has stated that the Ship Tease moments within the first film - specifically the dance scene, which is not in the book - were added in order to have something for the Harry/Hermione shippers.
Tastes Like Diabetes: The epilogue is this to its detractors. All it was missing was "And they all lived happily ever after." Then again, "All was well" is basically another version of that phrase. Someone on Livejournal created an animated icon that displayed the words "rocks fall everyone dies"; then, against a fluffy cloud background: All Was Well.
Trapped by Mountain Lions: The main storyline falls victim to this, which is the only point of view given after the first chapter. Obviously, Harry, Ron, and Hermione hunting down the Horcruxes is a big deal, but with its difficult pacing and long stretches of Dumbledore backstory only tangentially necessary to the plot, it sort of falls short compared to Voldemort having taken over the Ministry of Magic and NevillerunningLa Résistanceinside the school, which the Golden Trio (and thus the readers) only hear about secondhand.
Many fans were upset that Neville doesn't get to avenge his parents by being the one to kill Bellatrix. Though others counter that he does get his own big moment by killing Nagini, so that would have been overkill.
The resolution of Wormtail's plot smacks of this. Dumbledore was very on-the-nose about life debts at the end of Prisoner of Azkaban, but instead there's a brief moment where Wormtail shows up unexpectedly, hesitates in killing the heroes, and then dies as a result, barely warranting a mention after this. Any expectations of a redeeming himself or revealing some hidden depths, (or even, as some speculated, him telling Harry how to destroy the horcrux in him without killing himself) evaporate in half a page. Made even worse considering previous books. Goblet of Fire showed him trying to convince Voldemort to use someone other than Harry for his resurrection, insisting it's out of pragmatism rather than concern for Harry, and Half-Blood Prince showed him hanging out at Snape's, who explicitly mentions being mystified by Wormtail's new habit of listening at keyholes. While that could have set up another direction entirely for Wormtail, depending on what you read into it, it went nowhere.
The fact that Snape and Harry never actually confront each other is a huge anti-climax for many readers. Mostly because Snape only has very few scenes in the book, admittedly major ones, and he dies before he and Harry could talk about his true loyalties and feelings for his mother, nor could Snape apologize for giving the prophecy to Voldemort and treating him horribly over the years. This made the epilogue where Harry names his middle-child partly after Snape feel unearned. Likewise, Snape dying pathetically despite being established in Book 6 as a powerful wizard and proving his mettle briefly in Book 7 makes fans lament that there couldn't be a proper wizard's duel between Voldemort and Snape, with the final book unable to top the Dumbledore-Voldemort showdown of Book 5.
The entire R.A.B./Regulus Black reveal feels like this, mostly because the pay-off for the entire subplot is Kreacher's redemption, which to some readers feels unearned and wasted on a minor character at the expense of Sirius, a major fan-favourite of Books 3-5 with Hermione's dubious equivalency of Sirius and Voldemort's treatment of house-elves comes across as an awkward Author Tract and Informed Wrongness, and the fact that Regulus' revolt against Voldemort amounted to a giant Shoot the Shaggy Dog since he died without doing anything useful, giving the Horcrux to someone who had no idea or skills to destroy it. Voldemort never found that taunting message or knew that it was Regulus who duped him. The fact that Sirius died before knowing about Regulus only makes it feel more of a waste since the revelation about Regulus' allegiances does not have an impact on the character for whom it would have made the greatest difference. Likewise, the Horcrux, Slytherin's locket, keeps passing through multiple hands throughout the first part of the book which leads to the much-derided camping sequence, until the trio meet Xeno Lovegood and get taken to Malfoy Manor (where Harry stumbles on to the next plot-relevant object), meaning that RAB exists largely to service a weak aesop, add conflict, and bring in more padding.
The book often suggests more interesting storylines and Character Development among supporting characters (Ron and Neville) than Harry himself. Ron, for instance, starts out loyal, Took a Level in Jerkass and abandoned Harry and Hermione but eventually comes to rescue Harry in his moment of need and likewise learns first-hand a number of observations of the Wizarding World under Death Eater occupation. Many felt that it would have been more interesting to see that then have it told to us, especially because Ron has a sizable and vocal number of fans who dislike him. Likewise, a number of fans argue that Neville's Dumbledore's Army arc which comes in a big way in the end, and involves him befriending (entirely on his own) Aberforth Dumbledore and building a network with the wider resistance should have been seen more on-screen with many wishing that Rowling (who started the novel with a scene from Snape's POV and did so briefly in earlier books) took a similar approach with Neville and Ron, or even Ginny, Hermione and Luna.
In this book, the Death Eaters become a legitimate nation-wide threat and take over the Wizarding Government, unleashing a campaign of terror against Muggle-born and Muggles. This is the perfect territory for any The Unmasqued World scenarios, especially because the Death Eaters want Muggles to live in terror so they most likely wouldn't be using the enchantments designed to preserve The Masquerade anyway. Muggle Society could have also helped the good wizards by providing resources. However, the Death Eater's coup is still not treated as good enough of a reason for the Order of the Phoenix, the Golden Trio, or any of the other wizards to at least warn the non-wizarding population about the danger to let them know what's going on and give them a chance to protect themselves. We're also never given any indication of what Muggle Society went through during Voldemort's year-long control of the Ministry or even if The Masquerade was threatened because of the Death Eaters.
Unintentionally Unsympathetic: The fact that none of the good wizards bother to break The Masquerade to help the Muggles protect themselves from the Death Eaters makes it look like they value their isolationism over the fate of the world even when a world-ending threat will destroy both societies. It also further establishes the belief most wizards hold that Muggles aren't remotely intelligent enough to even defend themselves from anything even though the wizards refuse to let them know magic exists.
Pius Thicknesse is one of the biggest woobies of the series. His life's goal was to be Minister of Magic — but he took it for granted that he would be himself if/when he made it. Plus the sheer evil he was made to participate in.
The Gringott's Dragon, especially in the movie. The poor little thing is extremely old, somewhat weak, and almost blind, and by the time it escapes it seems to be absolutely astonished that it's finally free. The movie drives the point home by having it actually struggle to stay aloft at first, implying that it doesn't get out much (read: at all).
Aberforth and Ariana Dumbledore. Don't believe us? Read "The Missing Mirror" again.
The Grey Lady. In the book she comes off as self-obsessed, brooding forever on the wrongs that were done to her. The film version of the Grey Lady has anger as well as sorrow - anger at Tom Riddle for "defiling [the diadem] with Dark Magic," and it seems she regrets the wrongs that she did.
In the book, Gellert Grindelwald refused to tell Voldemort the location of the Elder Wand, but in the film he gives it up, smirking and laughing. Was he laughing at the thought of getting one over Dumbledore for one last time, or did he know that Voldemort would never be able to master the Wand?
Angst? What Angst?: Ron and Harry are pretty nonchalant in Malfoy Manor when they are shoved in a basement and their best friend is being tortured, with Lucius intending to summon Voldemort once they ascertain if his Horcrux is still untouched. In the book, Ron absolutely flips out when he hears Hermione screaming in pain, but in the movie he seems mildly alarmed. Hermione too seems to be merely a bit shaken after being tortured and having 'Mudblood' carved into her arm.
Anti-Climax Boss: Averted with Voldemort, whose climactic showdown with Harry takes the two on an epic battle all across the castle; this is a notable contrast to the book, in which Voldemort is defeated with much less fanfare. Played straight with Bellatrix, though, who is taken out rather easily for a dreaded Hero Killer.
Author's Saving Throw: The film cuts out the extended war of words between Harry and Voldemort right before their final duel. It instead makes the scene more action packed and climactic.
Award Snub: Considering that Deathly Hallows: Part 2 was not only the highest grossing movie of the year, but the highest grossing installment of what was then the highest grossing franchise, and one of the highest grossing films of all time, in addition to being one of the most critically acclaimed films of the year, would have at least garnered a few token nominations at the Academy Awards, especially since the Academy had made the decision several years earlier to expand the Best Picture field to ten films. But no, just three nominations in the technical department which it lost in all of them (the worst being losing Best Makeup to The Iron Lady!) It is also suspicious that before the nomination process began, the Academy again ammended its rules that up to 10 films could be nominated for Best Picture, if they gathered a certain percentage of votes. Coincidentally, only nine films were nominated, which prompted much outrage from both fans and film critics.note If you were wondering, these were Warner Bros. submissions to the Academy: Best Picture, Best Director - David Yates, Best Adapted Screenplay - Steven Kloves, Best Actor - Daniel Radcliffe and Rupert Grint, Best Actress - Emma Watson, Best Supporting Actor - Ralph Fiennes and Alan Rickman, Best Supporting Actress - Helena Bonham Carter and Maggie Smith.
The Harry/Hermione dance scene. It's often attacked for its awkwardness and for being pointless pandering to Harry/Hermione shippers. But there are just as many who love the scene for showing the deep bond between them, and providing a much-needed Heartwarming Moment after Ron's departure.
Splitting the film into two parts also divided fans. Some felt that the move allowed for better character development and a more faithful adaptation of the book. Others thought the decision was just another way for Warner Brothers to milk the franchise and the two movies could've been condensed.
Fanon Discontinuity: A lot of fans like to ignore Lavender's death, even if J.K later made it canon.
Harsher in Hindsight: Seamus joking that Ginny has "plenty of [brothers]" becomes this when Fred dies later that night in battle.
Padding: This criticism was placed against the first part of the film. While it keeps pace with the book, the first half of the book could have been compressed easily, resulting in what many find a tedious movie, sometimes mocked as Harry Potter Goes Camping.
Numerous characters are Demoted to Extra despite the film more or less revolving solely around the final battle, the reason why the book was split into two movies. The most notable example is perhaps Professor Sprout, whose return to the films was hyped up only for her to get a few silent cameos, disappointing many who have missed the character and her actress.
Ginny does not get a single moment during the battle of Hogwarts in heavy contrast to Neville, Dean, Seamus, McGonagall, Molly, Hagrid etc. She's only seen in group shots aside from giving Harry a Big Damn Kiss, further putting her in the role of Satellite Love Interest.
The film's version of The Tale of the Three Brothers has beautiful animation.
The (first half of the) Final Battle at Hogwarts is portrayed incredibly well, with Awesome Music, loads of CGI, and all sorts of spells flying back and forth. Not to mention the statues, the acromantulas...