Petunia suggests bringing Harry to the zoo on Dudley's birthday but leaving him in the car when they can't find a babysitter. Funny in 1997, but less so in the 2010s, with highly publicized cases and increased awareness of this being dangerous and potentially fatal to children.
One moment that could be this or (even more) Hilarious in Hindsight is the passage in the Christmas chapter describing the Weasley Twins enchanting snowballs to bounce off the back of Professor Quirell's turban after you've read the end of the book and discovered what he's keeping under there.
Another passage involves Hagrid forbidding Harry to buy a solid-gold cauldron. Imagine if Harry did buy that kind of cauldron, y'know, for Potions class...
Dumbledore, when asked by Harry about what he sees in the Mirror of Erised, amusingly replies that he sees himself holding a new pair of socks. Word Of God after the seventh book has confirmed that he was lying; what he actually sees is his dead sister Ariana alive and well, and his estranged brother Aberforth having forgiven him.
Rereading the opening of this book with Dumbledore telling McGonagall about how little they have had to celebrate for 11 years... and with the knowledge of how bad things were under Voldemort was when he was in charge in Deathly Hallows...
Harry's thoughts about the Dursleys taking all of his Gringotts money had they known about it becomes this with the reveal that one of the reasons uncle Vernon started disliking wizards is that; when Lily arranged a meeting between their families, Vernon took the tales of James' usual wizard routine (like moving around in flying brooms as opposed to cars) as wizards living off government help just to have James' casually tell him that his family is in general quite rich and goes as far as to explain wizard currency (which involves ACTUAL gold and silver coins) and mention his own (later Harry's) Gringotts' vault. To sink it further, James wasn't even trying to be malicious about it but Vernon still took it badly...
When Vernon is forced to admit he knew about Harry's magic powers, he dismisses it as "nothing a good beating wouldn't have cured". In Fantastic Beasts and Where to Find Them, we learn that using corporal punishment to suppress a young wizard's magic is a really, really bad idea.
Hilarious in Hindsight: Fred and George's winter prank on Quirrel becomes much funnier after the reveal at the end.
"Wingardium Leviosa!" and "It's Levi-OH-sa, not Levio-saaah."
"No post on Sundays!"
Moe: Watch the film and gush at the cuteness of Hermione, Harry and Ron!
Never Live It Down: A change from the book to the film is the Devils Snare scene. Fans erupted with fury at the fact that Hermione keeps her head and Ron panics, while it's the other way around in the book. But with the film removing Hermione solving the logic puzzle, it effectively leaves her without a Crowning Moment of Awesome in the lead up to the stone. The film also drops Hermione and Ron's parts in the flying key scene (they helped Harry catch it in the book, Harry does it on his own in the film). What's more is that the chess scene is still intact and Hermione does freak out here like she does in the book, and has to be reassured by Harry.
One-Scene Wonder: As expected with this franchise, but the literal examples are John Hurt as Ollivander and John Cleese as Nearly Headless Nick.
Ship-to-Ship Combat: Probably few people realized just how big and vicious of a battle this book would begin among its fans, most prominently among Harry/Hermione and Ron/Hermione shippers.
Slow-Paced Beginning: Notably, It takes about a third of the novel before Harry gets to Hogwarts. And quite a few children readers were turned off by the first few chapters for being too slow-paced; not to mention most of Rowling's dry satire on middle-class insecurities flying straight over their heads.
Squick: Dumbledore's experience with the Every Flavor Beans. He lost his liking to them a long time ago because of a vomit-flavored one, and when we see him actually try another, it was earwax-flavored.
In the film, Hermione says this word-for-word when Harry goes flying after Draco and the Remembrall.
Hagrid takes Harry, Ron (only in the film), Neville (in the book), Hermione and Malfoy into the Forbidden Forest to look for a Unicorn Murderer for detention. Harmful to Minors does not begin to describe Hagrid. Then again, he didn't really know what to expect.
Vernon insults magic and threatens Harry in front of Hagrid, a half-giant with super strength, who happens to be from the wizarding world. It's a good thing Hagrid isn't malicious, or Vernon would've probably ended up with a broken neck, or worse.
In the film the trio choose to visit Hagrid at night to talk about the Philosopher's Stone. They have no reason to do so, since they can easily visit him during the day. What's more is that they don't bring the Invisibility Cloak with them. This part is somewhat true to the book, as Harry and Hermione mistakenly leave it on the Astronomy Tower and walk right into Filch.
The Woobie: Harry is this at the beginning for living with family members who are abusive towards him.