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In the fifth book, Hermione tries to convince Umbridge that Dumbledore has a secret weapon hidden in the forest. She pretends to be crying while she does so. Harry notices she has her hands covering her eyes so as not to show she isn't actually crying. Except this convincing of Umbridge goes on for about three pages. And you realise Hermione was doing this the entire time with her hands over her eyes pretending to cry dramatically - and it becomes hilarious.
JK Rowling has been said to overdo it with some of Snape's dramatic moments. Remember "I, THE HALF BLOOD PRINCE?" or "Look... at... me..."
Harry's tantrum in Dumbledore's office in Order of the Phoenix was a strange variant of this trope. Over-the-top and horribly melodramatic and immature and hilarious on re-reads, when the reader's shock had worn off? Yes. But then you realize all the melodrama is in Harry's behavior, meaning that he's so heartbroken he can't even stop himself from acting embarrassingly.
The tantrum Harry throws at Ron and Hermione earlier in the book is a much straighter example. It's essentially a couple of pages where all of Harry's lines are in block capitals to show he's angry. Fans have nicknamed that part 'Caps Lock Harry'.
In-universe, Harry thinks pretty much any time the Dursleys show emotion is Narmy, mostly involving Petunia and Dudley. Whenever the Dursleys show affection for each other, expect Harry to "suppress the urge to laugh".
The epilogue was Tastes Like Diabetes for some. Regarding that epilogue, it seems the real issue is that it was actually the first thing Rowling wrote of the series, and she didn't do enough when the time came to edit it to match the way her writing style or the tone of the series had evolved.
From the audio books, Stephen Fry chose to give Tonks a thick Yorkshire accent due to her using the Oop North expression "wotcher". It was passable in the fifth book but don't forget the amount of dramatic scenes the character has in the sixth. Tonks's Anguished Declaration of Love at the end is made hilarious because of the accent Fry is using.
From Prisoner of Azkaban - "HE WAS THEIR FRIEND!"
Same movie, the punching scene. For some reason the writers decided to change the dialogue from the book. They thought "you foul loathsome evil little cockroach!" would be better. The stupidity of the line, coupled with awful acting from Emma Watson and Tom Felton just makes the scene ridiculous. Plus the aforementioned line change comes across as Gosh Darn It to Heck!.
Again from POA: "I DID MY WAITING! TWELVE YEARS OF IT! IN AZKABAN!"
The fourth movie, Harry's putdown to Malfoy that enrages him so much? "Your father is vile and cruel"
I thought the thing that set him off was "you're just pathetic."
Hermione's ridiculously over the top reaction to Moody teaching the Unforgivable Curses. Especially her tearful shaking of her head when she refuses to say Avada Kedavra.
When Harry's wand breaks, Ron brings him a new one, saying "Ten inches, nothing special." That's what she said.
That bit in the beginning of HP and the Deathly Hallows, when Voldemort tells Nagini to eat the body of the teacher. Voldy, do you know how long it takes for snakes to swallow things? Nagini lunges at the camera, fangs extended... and all I can picture is her then settling down, after that terrifying LUNGE, to slowly wedge the body little by little into her mouth over the course of the next few hours.
In the third film, there's the scene in Trelawney's class when Harry gets "The Grim" in his tea cup. Some kid starts to explain it: "It’s an omen… *dramatic pause* the WORST omen… *pause so dramatic that anyone in the vicinity had this been real life would have burst out laughing* of DEATH." *cue dramatic music as we see the camera slowly zooming in on a ridiculously clearly defined dog in the tea leaves* It's obviously supposed to be foreboding, but I can't help but crack up every time I see it.
Also in the third film, when Malfoy and gang run into Ron and Hermione in Hogsmeade. Malfoy reacts stupidly over the top to a casual putdown from Hermione and Tom Felton's delivery of "how dare you talk to me!"...
Are we not going to talk about Voldemort's hug to Draco in Deathly Hallows Part 2. The hug scene is so infamously Narmy that not only have both Tom Felton and Ralph Fiennes called it funny, but it has also spawned such things like "On a scale of 1 to Voldemort, how awkward are your hugs?"