In the film, when Harry jumps through the wall of fire to pursue Bellatrix, Lupin is the first of the adults to run forward. Is he concerned for Harry's safety, or is he going after the woman who killed his best friend? Probably both.
Merope dying after her son is born. Harry wonders why she didn't try to stay alive for the sake of her child. Perhaps she came to resent the baby growing in her womb? As a remnant of the man she had loved who rejected her, perhaps she had no plans to raise the child at all since she actively went into an orphanage to give birth.
Author's Saving Throw: This book contains two subplots where Love Potions are used in situations that are definitely notDouble Standard: Rape, Female on Male, in a possible response to criticism that said Potions are basically G-rated date rape drugs marketed to Potterverse witches (for added contrast, Fred and George do exactly that in this book and it's just as creepy as it sounds). In the present day, Romilda Vane is seen as a disturbed stalker after she tries to send Harry a box of chocolates that were spiked with Love Potion; and in the backstory, Merope Gaunt, Voldemort's mother, kept Tom Riddle under various enchantments, including Love Potion, so he wouldn't leave her.
Big-Lipped Alligator Moment: The Burrow-burning and Bellatrix's appearance in The Movie; Word of God says The Burrow attack was to bring the threat of the Death Eaters more close to home to the characters than Hermione reading about it in the papers, but it still arrives and leaves without another word in true BLAM fashion.
The book that ignited the greatest quantity of flame wars on forums. Especially over the Ship Sinking.
Minor one but Hermione's crying after Ron kisses Lavender. Some found it Narm, while others had a She Really Can Act reaction towards Emma Watson.
Lavender being played by a white actress was another one, mostly since she'd been entirely in the background with no lines in the previous films. Some felt it was pointless to change her back to white, while others felt it was more in line with the booknote When Ron and Lavender are making out, Harry mentions that it's hard to tell whose hands were whose, implying Lavender to be white.
Captain Obvious Reveal: Snape being the Half-Blood Prince surprised nobody. The first clue was Snape getting promoted to Defense Against the Dark Arts teacher, which is a dead giveaway that a character is going to be important in a Harry Potter book. The second, more glaring hint was that the HBP was good at potions.
Harry has always had a Screw the Rules, I'm Doing What's Right! but in Half-Blood Prince, he indulges in rule-breaking for the sheer heck of it. He cheats at Potions and commits what in real-life would be academic plagiarism, he uses spells on random targets (including Filch's cat), unleashes Sectumsempra on Draco despite knowing nothing about the spell (Although, in his defense, he did seem horrified at the result). There's also the fact that he gets to be Quidditch captain despite being suspended off the winning team the previous year out of his poor temperament and yet doesn't fully defend Ron (the Keeper who more or less won the final the previous year and came through on a big game), and once again doesn't take his leadership responsibilities seriously, misses the final game again due to detention and the team wins without him again.
Designated Villain: Cormac. He may be a jerk, but he suffers through a lot of Disproportionate Retribution, like having his Quidditch try-out rigged or having a month of detention for accidentally vomiting on Snape. Though, in the book, he's even worse, Hermione is called out on rigging his tryouts by Harry, and the match against Hufflepuff proves that the team is much better off with Ron as Keeper.
Double Standard: Abuse, Female on Male: To say the least of the infamous bird attack scene. Many a Ron the Death Eater fanfics will have Ron attack Hermione in a fit of jealousy. Hermione attacks Ron in canon, and it is for the most part swept under the rug. note While both sides are portrayed as being petty, Hermione is still never called out on for resorting to violence. The film at least softens this, so that Ron isn't actually hurt with the birds like he is in the books.
Draco in Leather Pants: Hey Draco, that's a nice suit you got there. Of course, it helps that Draco starts to become a lot more pitiable by the end of the book, what with it being revealed that he's terrified of dying and/or causing the death of his family.
Fan Nickname: Disappointment in the movie caused some fans to nickname it "Harry Potter and the Half-Baked Script".
Foe Yay: Thank God Harry's obvious obsession with Draco was eaten by the giant chest monster, who decided to save the appearances just in time. Note that the word 'obsessed' is used in-story by both the narrator and a rather-concerned Ron.
"Funny Aneurysm" Moment: After Lavender runs out of the infirmary in tears, Dumbledore muses "Ah, to be young and feel love's keen sting." Rowling later mentioned that Dumbledore had a crush on Grindelwald and that a fight where Grindelwald was trying to torture Aberforth eventually led to their sister's death. Dumbledore also held off going after Grindelwald due to his lingering feelings for him until it was obvious he was too powerful for Dumbledore to ignore.
Also Harsher in Hindsight: Dumbledore's "Oh, to be young and to feel love's keen sting" line in the infirmary upon Lavender hearing a bedridden, unconscious Ron call out to Hermione. In the backstory provided by Rowling, Dumbledore was in love with Grindelwald, who left the country after a fight Grindelwald had with Dumbledore's brother led to the death of their sister. It was also Dumbledore who had to put an end to Grindelwald's reign of terror.
In the first book, Ron sees in the Mirror of Erised himself as Head Boy and holding the Quidditch Cup. He manages both (almost, as he became a Prefect) in this book and film. Also a little harsh considering that Ron never attends his last year at Hogwarts, meaning that there's a chance that he could have qualified for Head Boy but still couldn't take the position.
In the movie, Bellatrix Lestrange's actress was pregnant and had to hide it during filming. Come Cursed Child, it's revealed that her character is in fact the mother of Voldemort's child.
In a related line from the books, Bellatrix tells her sister Narcissa that if Bella ever had sons, she would gladly give them up in service to the Dark Lord. Bellatrix and Voldemort's daughter Delphi ends up with roughly the same fate.
At the end of the book, Bill Weasley is described as bearing "a distinct resemblance to Mad-Eye Moody" after being attacked by Fenir Greyback. In the Harry Potter and the Deathly Hallows movie, Brendan Gleeson's son Domhnall Gleeson played Bill Weasley.
Outside of the Ship-to-Ship Combat, words can not describe the level of backlash fangirls displayed when Blaise Zabini was revealed to be, wait for it, black. Which was followed by "Wait, Blaise is a DUDE!?" Pretty much making thousands of Female Blaise fics void (or just a different genre now).
There were also a lot of complaints from non-shippers who were bothered by the book's focus on romance. Probably at least in part a Hype Backlash due to the massive focus those subplots got in the eyes of the rest of the fandom, but even so, it's a little unreasonable that they expected a book about 16-year-olds at a boarding school not to discuss dating and romance.
The trailer containing shots of the Burrow burning provoked a lot of complaints about an Adaptation-Induced Plot Hole for the seventh book - as the opening part of it takes place there. They seemed to ignore the six month difference between the events.
Jerkass Woobie: Morfin Gaunt. While he is established to be a fairly nasty wizard, it becomes hard not to feel sorry for him after he loses his father to Azkaban and Voldemort frames him for three of the murders he committed. Dumbledore himself lampshades it.
Similarly, "SNAPE KILLS DUMBLEDORE ON PAGE 606/596." (page 606 is commonly used as it has Harry explicitly saying "Snape killed Dumbledore," but the actual death occurs on 596. Or 566/556, depending on which edition you're reading.)
Amusingly, someone changed the Wikipedia page for the book to spoiler the ending. The IP address traced back to the HQ of the Minnesota Republican Party.
Moral Event Horizon: Draco nearly crosses it when he tries to use the Cruciatus Curse on Harry in the bathroom. Subverted when he tries to kill Dumbledore because he doesn't have the nerve to do it and is only trying out of fear of Voldemort killing him or his parents.
Misaimed Fandom: Some fans, even some editors on the Wiki, tend to sympathize with Merope too much claiming her using a Love Potion on Tom Riddle was justified; even going so far as to claim Tom Riddle was the bad one, pointing to a few scenes where he's described as a snob (this is ignoring his Pet the Dog moments like taking care of his elderly parents), and him leaving a pregnant Merope and never even bothering to find out what happened to his child, never mind the fact that he was mind-controlled and magically raped, and didn't consent to the relationship in the first place. This would not exist if it were Tom who did this to Merope, mind you.
Narm: Harry's romantic interest in Ginny is generally fraught with silliness, like his fears that her brothers will suddenly hate him, but it's the monster living inside his chest that takes the cake, especially when it starts roaring.
Poor, poor Ginny. There's no evidence that she goes further than kissing with anyone, and with a grand total of two boys before Harry, and yet the bashers are all too happy to label her the "Hogwarts whore". Especially amusing as Ginny has to put up with the same accusations from her brothers in the very same book. She swiftly shuts them up.
Lupin and Tonks. Harry sees Tonks upset several times and thinks it's over Sirius dying, and that maybe she was even in love with him. Then the climax reveals that Tonks is deeply in love with and wants to marry Lupin. They proceed to do so despite having no interaction on camera before this (although eagle eyed readers will notice that they tend to work together a lot in Order of the Phoenix). As the books are primarily written from Harry's perspective, it's somewhat understandable that plenty of major events can happen off-camera simply because Harry can't be present for every single storyline. At the same time, the reader can still feel left out as the key parts of the Romance Arc never actually happen in front of them either.
Take That, Scrappy!: Dumbledore coldly confronting the Dursleys for all the abuse they put Harry through over the years, and pointing out that the best thing that they ever did for Harry was to guarentee that he didn't grow up to become a horrifically Spoiled Brat like Dudley.
Hermione during her feud with Ron. While what Ron did may have been a low blow, Hermione reacts to it with complete immaturity by assaulting Ron with birds. Her inability to take the high road was only Hand Waved, as mentioned in the Double Standard: Abuse, Female on Male entry. Harry likewise seems to lose any sympathy for her when she invites Cormac MacLaggen to the Christmas party, solely to annoy Ron.
Harry Potter's actions in the book are also seen by many to be incredibly dubious. There's the fact that Harry uses the Prince's book to more or less cheat at Potions, which in real life would amount to plagiarism and would immediately nullify his grades were a real-life student used similar methods.