Harry Potter and the Philosopher's Stone
Much of the first chapter talks about wizards celebrating the fall of Voldemort all day long, yet Harry's family was attacked at night; unless it took Hagrid 24 hours to get Harry from his parents' ruined house to Little Whinging, this seemingly does not make sense. A possible explanation, however, is that it does not say when Hagrid pulled him from the house. It could have been a full day until they could respond, or he could have kept Harry with him until night, when he could easily fly under cover of darkness to Little Whinging. He could have also gone to get the baby Harry checked over to see if he had been harmed. The day, however, is November 1st, the day after Voldemort's downfall, so the celebrations are happening after his downfall the evening prior.
The beginning of the story takes place on a "dull grey Tuesday." In 1981, however, Oct 31 was on a Saturday, placing the celebrations on Nov 3. This makes what the TV weatherman wrong as he admonished people that Bonfire Night was the next week. In fact, it would have been in two days.
The boa constrictor at the Zoo winked at Harry during their conversation. And while snakes do have eyelids, it is impossible for snakes to blink like the boa constrictor did to Harry, their eyes being protected by transparent scales.
Harry's 11th birthday, 31 July, was a Tuesday. But in 1991 in the real world, that date was a Wednesday.
When Harry and Hagrid are leaving the little shack out in the middle of the sea, they used the rowboat that the Dursleys' had borrowed in order to leave the rock. This leaves the Dursleys with no way to get back to shore, yet there is no mention of retrieving them, and it is implied Harry goes straight back to Privet Drive after his shopping at Diagon Alley.
On Harry's Hogwarts list of school supplies, "1 Wand" is listed twice, as both the first and last item on the list. (Rectified in later editions)
It states in the Hogwarts letter that you can bring an owl, toad, or cat. Ron brings his rat Scabbers. It is possible, however, that the rules are leniently enforced, or despite those specific animals being listed, there is no specific prohibition against other types of animals.
The air in the tunnels to the Gringotts vaults is described as getting colder as they go deeper. This only happens very near the surface, and only when the weather has been relatively warm. Once you get below the level where the temperature depends heavily on the seasons at the surface, the geothermal gradient takes over, and it gets hotter as you go deeper. In the world's deepest mines, the miners can only work for a short time before being brought back up, because of the risk of heat stroke.
Much importance is placed on the theme of wandlore and wand ownership throughout the series, so it seems odd in retrospect that, during Harry's first meeting with Draco Malfoy in Madam Malkin's, he should mention that his mother is away looking at wands. Indeed, it is just as odd to hear that Ron has been equipped with Charlie's old wand. It is mentioned, however, in Deathly Hallows and on Pottermore that there are old superstitions about wands and what kind of people are chosen by certain wands and so it's possible that Narcissa Malfoy had certain beliefs along those lines. As for Ron, his parents probably could not afford to buy him a new wand at the time. This was probably meant to show that his family was not very well off in terms of wealth. Even though it is better to have your own wand, a person can still do magic with another person's wand.
When Harry is talking to Hagrid about his first meeting with Draco Malfoy, Hagrid says that "there's not a single witch or wizard who went bad that wasn't in Slytherin." At the point when this conversation takes place, Hagrid (along with the rest of the wizarding world) would have been under the impression that Sirius Black, a Gryffindor, was responsible for betraying Harry's parents. This, however, is potentially simply an instance of the literary device of hyperbole ("the use of a statment to evoke strong feelings or create a strong impression, but not meant to be taken literally"), rather than ever being intended as an accurate historical statement, even if one ignores the fact that there are known to be dark wizards from other schools, such as Grindelwald.
When Harry returns from Diagon Alley to Little Whinging, he takes a train from Paddington station. But Little Whinging is in Surrey, south of London, so he should have travelled from Victoria or Waterloo; trains from Paddington head to the west.
Hermione Granger says that she has been practising spells at home, but this is against the Decree for the Reasonable Restriction of Underage Sorcery, which normally results in a warning letter. Hermione, however, was Muggle-born, and had not yet started school, so it is possible that Muggle-born children are not informed of these rules prior to starting school. Alternatively, it is also possible that she practiced the wrist movements one does when executing spells, or the correct pronunciation of the spells, rather than the actual execution of the spells.
While being Sorted into houses, Harry looks up at the Sorting Hat's stool, and there are only three people left to be sorted. Professor McGonagall then calls out the names of four more students. (This is only in the US edition - in the UK edition, Dean Thomas's name is omitted. Rectified in later editions).
When Harry first met Nicholas de Mimsy-Porpington, the ghost says he had not eaten in nearly 400 years. Yet he had, at the time, already been dead for 99 years longer than that. (Rectified in later editions).
Harry buys the book One Thousand Magical Herbs and Fungi, but later, he looks up "dittany" in One Hundred Magical Herbs and Fungi. (Rectified in later editions).
Fred Weasley says that Gryffindor has not won the Quidditch Cup since Charlie left. According to J.K. Rowling, Charlie is three years older than Percy who is in his fifth year, meaning that Charlie would have left the year before. So basically Fred is saying that Gryffindor has not won since last year. Professor McGonagall also said that Gryffindor was flattened in the last game between Gryffindor and Slytherin. While this is not an outright mistake, it is still somewhat confusing. It is possible that, like Harry, Charlie was an excellent Quidditch player, but still only won the Cup on one or two occasions due to other circumstances. Another confusing part is that readers confuse the Quidditch Cup with the House Cup. At this time Slytherin was on a six-year winning streak with the House Cup, not the Quidditch Cup.
When Harry tells Ron about him getting selected as a seeker for Gryffindor they both agree that he — Harry — is the youngest house player in about a century. However Katie Bell who was now in her second year also played for Gryffindor got selected in her first year because it is nowhere mentioned that Katie Bell was a new player, it is also not mentioned that Wood held tryouts for chasers, so Harry could not be the youngest player in a century.
During the Hallowe'en feast, when the troll is let in, Professor Dumbledore sends the students to their dormitories. However, the troll is said to be in the dungeons, and that is also where the Slytherin dormitories are, meaning Dumbledore has put them directly in harm's way by sending them there. He did, however, send the sixth and seventh years with them and given that the Trio were able to handle a single troll on their own, one would not be able to overcome that many wizards.
During the match against the giant chess set, Ron says, "Well Harry, you take the place of that bishop, and Hermione, you go next to him instead of that castle." But the castle is next to the knight, not the bishop. It is, however, possible he meant "next to him" to mean "on the same side of the king and queen."
At the beginning of the chess match, Ron is the knight in the giant chess set. However, it later says that he had to "move ahead one" so the Queen could take him, opening the path for Harry to checkmate the king. If he was a knight, he could only move 1) ahead one, sideways two, or 2) ahead two, sideways one, like an "L." It is possible this meant "one" as in a single move, but it is still an odd construction regardless.
Flint is referred to as a burly 6th year, but he is in book 3, meaning either his age was forgotten by J. K. Rowling, or he dropped a year. It was later confirmed by Rowling that Flint failed his N.E.W.T.s and had to repeat his seventh year.
On Page 173 of the UK paperback, in "Norbert the Norwegian Ridgeback", it says "Wednesday night found Harry and Hermione sitting alone in the common room, long after everyone else had gone to bed. The clock had just chimed midnight...." However in Chapter 8, "The Potions Master", it is mentioned that Astonomy classes take place every Wednesday at midnight. Therefore, they are presumably skipping class, and the other students should not be asleep. (Norbert's departure is changed to Saturday in later editions.)