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.
Ass Pull: The revelation that the reason Harry's Invisibility Cloak never tears or falls into disarray is because it is one of the eponymous Deathly Hallows is a clever inversion of this, 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 attentive readers will notice Mad-Eye Moody talking about having to borrow a fellow Order member's Invisibility Cloak because his was starting to wear off. The average reader would wave it off as a continuity error, but it it becomes an important plot device in the final book.
Similarly, Wand Lore seems like this until one remembers the very first book. "The Wand Chooses the Wizard" indeed.
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.
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.
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.
She casts a corporeal Patronus, capable of holding back several Dementors at once, while actively taking joy in the Inferred Holocaust of Muggle-borns and wearing the Horcrux locket around her neck. Repeat: the happy thought she is using to summon her Patronus is the mental image of people getting their souls ripped from their bodies by beings that embody fear and despression. It's around this point that Voldemort starts to sound like The Dragon in comparison.
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.
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.
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 worse 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.
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.
Jerkass Has a Point: Muriel Weasely mentions in Death Hallows that squibs were encouraged to settle down in Muggle's world, and views it as blessing for them, because they would live as lower class in Wizard's world. It's indeed true - just remember Argus Filch... though he himself chose such a fate. And Arabella Figg seems to be quite happy to live in Muggle's world.
Jerkass Woobie: 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.
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.
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.
"NOT MY DAUGHTER, YOU BITCH!!!!", which made Molly Weasley ascend to badass status.
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.
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.
Narm: In the book, Harry suggesting Voldemort try for some remorse. You can almost feel Harry thinking "this sounds really stupid, but I've gotta give him the chance."
The name of Harry's younger son: Albus Severus Potter. Actually, his other kids' names are pretty narmy as well — James Sirius and Lily Luna — but poor Albus got stuck with the worst one.
What Ron sees when he destroys the locket. Made even worse when you realize that the locket was probably telling him those exact same things, over and over, in the time before he ran away.
The scene where Harry uses the Cruciatus Curse on Amycus Carrow. Thankfully, this scene was omitted in the film.
Nagini using Bathilda Bagshot's hollowed-out corpse as a Meat Puppet. Harry even decided to spare Hermione the details of Nagini's arrival.
Another scene that was ultimately cut from the film was Peter Pettigrew being forced to strangle himself to death, rumor has it because the censors considered it so messed up they would've given the film an R rating otherwise. Unfortunately, this turned that villain into a Karma Houdini as no alternate death was written for him.
The Power of Love: Voldemort is universally recognized as the greatest Legilimens in the world, to the point that only the greatest Occlumens in the world (Snape) can resist him, and can ferret out lies without word or wand. Yet Narcissa Malfoy, a competent yet average witch, manages to lie to his face. At that moment, the only thing going through her mind is the love of her family, something Voldemort doesn't understand, doesn't believe in, and has no defense against.
J. K. Rowling chose to sink the Harry×Hermione shipthrice, the third time of which was in this book. First, in an interview about the possible pairing, Rowling said she didn't think they were suited for each other. Secondly in Half-Blood Prince, and then the internet exploded, with Harry/Hermione shipperssending out death threats, burning books, and swearing to abandon the series if she didn't change it. To drive the nail in the coffin, this book has Harry giving the "Like Brother and Sister" speech to Ron and Hermione, right after a scene in which Harry/Hermione appears as an evil illusion made by the locket-Horcrux that seems vaguely perverse and is intended entirely to enrage Ron into attacking Harry by using frequent Harmonian arguments, at that. Rowling has mentioned that Harry/Hermione make a better match in some ways than Ron/Hermione but it would never happen because Harry's true love is Ginny and Hermione would be miserable without Ron.
She also said Neville×Luna wasn't canon. On the other hand, the movie has Neville tell Harry he has to tell Luna how he feels. And There Was Much Rejoicing for the Neville×Luna shippers. Which was sunk again by Matthew Lewis (who plays Neville in the films), when he said the Neville×Luna relationship was ultimately only a "summer fling", and that they did eventually move on to their respective canon pairings. J. K. Rowling used to reject the ship but has stated that she grew fond of over time and could see it working out, which is why she approved of the aforementioned changes the film made.
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.
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...
They Wasted a Perfectly Good Plot: 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 book 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.
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.
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.
Snape, particularly as a kid and teenager where he keeps messing things up with Lily despite how hard he tries. However, certainly being unable to protect the woman you love and then protecting her child who was sired by and looks exactly like the man you hate (except the eyes) has got to be the rawer end of the deal.
Nearly every major plot point has somehow been a slap in the face for this guy, from Harry's appearance to Voldemort returning. He was the bane of Harry's existence, but, in the end, all you can feel is pity.
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)
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."
When Voldemort tries to break the barrier around Hogwarts, the camera zips into the elder wand showing it cracking under the stress. Thing is, the wand is never damaged and these cracks never come up again, even though a fair bit of CGI was used to illustrate it. The wand isn't working for Voldemort, but it isn't being damaged...
Ear Worm: The Nick Cave song ("O Children" from The Lyre of Orpheus) that plays while Harry and Hermoine are dancing in the tent.
Hilarious in Hindsight: The "NYEAH!!" scream of Voldemort that become famous thanks to the Part II trailer was actually heard once in Part 1, right after Harry's wand overpowers him in the aerial battle.
"NOT MY DAUGHTER, YOU BITCH!!!!", which just jarred with her kind and maternal (if somewhat shrew) personality.
On a scale of one to Lord Voldemort, how awkward are your hugs?
Voldemort's scream from the trailers: NYYYYYYYYYYYYYYYYEEEEEEEAAAAAAH!!!!!
Narm: 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 it may seem strange until one remembers what happen to him (in Quirrel's body) after he was defeated in the very first movie, he collapsed into ashes.
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.
Nightmare Fuel: In the movie, during Harry's tussle with Voldemort on the Hogwarts towers. The movie has Harry ride along with Voldermort as he tries to kill him around and through the building! And at one moment, the camera zooms in to show that the two characters' faces have merged! shudder
This effect brings up more questions - because this "merging" part occurs after Harry was already rid of the part of Voldemort which was connecting them. However, seeing as at the time Nagini was still alive, perhaps what was left of Voldemort's soul was trying to reclaim what it lost (and failed, because the piece in Harry was gone)...
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.
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.