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 Chekhov's Armory in the books. Harry takes the power of his Cloak for granted, simply assuming that all Invisibility Cloaks are just that awesome, but nothing about its status or unique qualities were mentioned or suggested even once.
The concept of Wand Lore. It's true that 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 dangerous to attempt, since its flames hunt down its victims. The fact that the object is a Horcrux that is mentioned as being Made of Indestructium 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?".
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.
Continuity Lock-Out: This book is the first to not introduce Harry's point of view section by his first and last name—previous books started out with calling him "Harry Potter". This one, once it gets to him, just says "Harry was bleeding."
It's not touched on much, but there are a lot of parallels between Draco Malfoy and Regulus Black. Both were raised in very old, pure-blood Slytherin families (actually the same family, technically - Regulus is actually Draco's first cousin, once removed through Draco's mother, Narcissa). Both parroted the pureblood supremacist ideals of their families for much of their lives. Both joined the Death Eaters at or around the age of 16 and were given special, specific assignments. Both came to realize that Evil Is Not a Toy and showed a desire to back out. And both have a huge part to do with Voldemort's defeat - Draco much more unknowingly.
Word of God says that Teddy Lupin's upbringing is said to mirror both Neville'snote Lost both parents to evil but is raised by a caring grandmother and Harry'snote taken in by a loving non-blood family and eventually falling in love with a member of that family.
Designated Hero: Snape in the scene where he tears off the photograph of Lily, James and Harry and keeps a part of Lily's message for himself. The way the scene is written indicates his great grief for Lily and his love for her, but his actions amount to posthumously disrespecting Lily's consent and her choices (which as Dumbledore insists is what defines us), namely taking her away from her happy marriage, and ignoring the fact that the letter with "Lots of Love" was addressed to Sirius and not him, and it's way too reminiscent of a Stalker With a Crush who steals stuff to build a Stalker Shrine than sincere grief.
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.
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.
Fanon Discontinuity: 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?". It doesn't apply only to shippers. Quite a few people disliked the Epilogue for its reputation of "Tastes Like Diabetes."
Also Molly killing Bellatrix, an act which many believed (and still believe) belonged rightfully to Neville.
Quite 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.
Reading the scenes in Chapter 8 that involve Elphias Doge (who is always refered to as 'Doge') is much more hilarious if one is familiar with the "Doge"meme
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 treat him in such fashion, and the idea that being cold and indifferent to someone makes one the same as the Big Bad in the eyes of Kreacher makes it come across as Disproportionate Retribution.
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.
Magnificent Bastard: Once you've come this far in the series, it's settled. This title doesn't belong to Voldemort, it goes to none other than Albus Dumbledore himself. Directly and indirectly, intentionally and otherwise, this one man, has been responsible for everything, everything, that has driven the history of the Potterverse from his time and beyond, fitting every characteristic trope almost all the time.
Memetic Badass: 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.
Memetic Mutation: "NOT MY DAUGHTER, YOU BITCH!!!!", which made Molly Weasley ascend to badass status.
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, even when Draco is dead set against it.
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.
Then again, "All was well." is basically another version of that phrase.
The Untwist: Wormtail's life debt to Harry isn't repaid out of gratitude or a Heel–Face Turn, but instead he's simply magically prevented from harming Harry at a critical moment. His "mercy" is completely accidental, and then triggers a You Have Failed Me failsafe...
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 Face–Heel Turn or some Hidden Depths evaporate in half a page. Made even worse considering the 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 feels 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 wizards 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 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 Coupon), meaning that RAB exists largely to service a weak aesop, Conflict Ball, and 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 than 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.
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 back story 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.
Values Dissonance: The entire Kreacher subplot involves Harry, The Hero, and Hermione, who had taken on Elf-Abolitionism in Book 4, to accept that Kreacher truly does have Happiness in Slavery and that Regulus Black is somehow vindicated for being a kind slave master as opposed to Sirius who didn't want to own any slaves and disliked being forced into a culture he had spent all his life rebelling. It's related to us by explaining that the House-Elves have Blue and Orange Morality which undercuts much of the intended aesops stated elsewhere, and which amounts to more or less normalizing slavery to the extent that Harry ends the series as Kreacher's slaveowner and the final lines (before the epilogue) has Harry muttering that he'd summon Kreacher to bring him a sandwich, which means that Harry kept Kreacher as a slave even after Voldemort fell.
What an Idiot: Yes, to safeguard the Horcrux, let's wear it at all times. It's not like we had proof that said Horcruxes can posses or damage the bearer (it actually tries to drown Harry at one point). And the feelings of paranoia and anger, plus the deterioration of friendship... normal. In the book, the trio knew that the Horcrux affected them but still carried it everywhere. Rather than, say putting it in Hermione's enchanted bag that can keep many things inside and shrinks down when needed to prevent it from being stolen, just as they were doing with all of their other supplies and equipment. Or putting it in Harry's magic pouch, which also has charms on it for security and safekeeping. They could have even simply worn the bag if carrying it was the problem.
It was mentioned that the Horcrux was immune to magic and Hermione's bag was at one point so messy that they have to use Accio to find something.
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 me? Read "The Missing Mirror" again.
Specific to the films (Parts 1 & 2)
Alternative Character Interpretation: 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.
Award Snub: Considering that Deathly Hallows Part 2 was not only the highest grossing movie of the year, but the highest grossing installment of 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 - Ralphe Fiennes and Alan Rickman, Best Supporting Actress - Helena Bonham Carter and Maggie Smith.
Some will certainly view the dance between Harry and Hermione after Ron's departure in the first film as such. Although it serves to illustrate what is never explicitly stated in the film: that Harry and Hermione are like siblings and will never be a romantic couple. Trying to show that with a scene that many (even non-shippers) saw as very romantic, was probably not the best way. Aside from romantic information, the scene serves to show how much shit has hit the fan and that no amount of having fun will lift their spirits and build Harry's confidence to take on his inevitable task.
Emma Watson has said she was quite weirded out by the scene, and even told the director "Are you sure you want to do this? It wasn't in the books, you know."
After he (seemingly) kills Harry during the climax of the second film, Voldemort approaches the school's defenders and offers them the chance to join the ranks of the Death Eaters. Draco, after hesitating, steps forward to join his parents, and is hugged by Voldemort on the way. Needless to say several theater houses were in stitches during that bit. This was probably intentional, considering Draco reacts like he's being hugged by, well, Voldemort (and is only doing it so he can leave with his parents) and Voldemort is pointing the Elder Wand at the other students the entire time
After being attacked by Voldemort, Neville wakes up and sees the sword with the derpiest possible expression.
Bellatrix and Voldemort dissolving into flakes for no apparent reason in the climax. Bellatrix was exploded after being petrified, that made some sense. Voldemort, not so much.
Voldemort's dissolving into ashes may seem strange until one remembers what happened to him (in Quirrel's body) after he was defeated in the very first movie, he collapsed into ashes. It still rings hollow with some who appreciated how the book described that for all his work towards immortality, his death comes with his corpse hitting the ground as a mortal man.
The scene towards the end where Lupin offers some slightly incoherent wisdom to Kingsley ("Who said that?" "Me.") comes off as just a bit silly.
"Come on, Tom... let's finish this the way we started it." (hugs him aggressively) "TOGETHER."
Worse is that the way he throws his arms around Voldemort's neck makes it look like he's about to plant a big kiss on him. Yuck.
Snape hugging Lily's dead body while baby Harry cries in the background is so over-dramatic it seems Narm-y to some viewers.
While "Dobby has no master!" was meant to be a Crowning Moment of Awesome, some felt it loses some of its drama when spoken in a high-pitched falsetto by a creature that looks like the love child of Vladimir Putin and Jar Jar Binks.
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.
In the film, while immensely satisfying to see at the same time, the Voldy-flakes produced as Voldemort disintegrates. In 3D.
"Nagini... dinner." Becomes Fridge Horror when you think about the digestive consequences, which may have been going through Draco's mind based on his expression.
To some the scene where Snape embraces Lily is filled with this. Book readers find it creepier since Snape in the novels never got any consent from Lily, and the scene of Snape hugging the body replaces an altogether more pathetic and less romantic scene of him tearing a family photo of the Potters, keeping only the Lily part of the photo. The new scene by juxtaposition evokes necrophilia.
They Wasted a Perfectly Good Character: 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.
They Wasted a Perfectly Good Plot: The movie almost completely omits the final battle in the Great Hall; all we see are some blurry bangs and flashes when Neville wakes up and Molly's very brief duel with Bellatrix.