Continuity mistake: At Nick's 500th deathday, which occurs on 31 October (clearly stated in the book), it says that he died in 1492. Therefore, the book takes place in 1992, and since Harry is currently 12 years old, he was born in 1980. Yet in the first book, Harry mentions that his 11th birthday was July 31st, on a Tuesday. That date was on a Wednesday in 1991, when he's supposed to have been born.
Continuity mistake: In the second book Harry, Ron and Hermione go to Nearly Headless Nick's 500th deathday party. However, in the first book at the start of term feast he says he has not eaten for nearly 400 years, which would imply that this is how long he has been dead. (Corrected to 500 years in newer editions.)
Continuity mistake: In the chapter 'The Rogue Bludger', Harry breaks his arm in the quidditch match against Slytherin (and then Lockhart debones it). During the night in the hospital wing Dumbledore and McGonagall bring up Colin Creevey who has been petrified. Dumbledore claims "Minerva found him on the stairs" but further down McGonagall says to Madam Pomfrey "but I shudder to think...if Albus hadn't been on the way downstairs for hot chocolate, who knows what might have...", implying that Dumbledore actually found him.
Other mistake: In the chapter "Dobby's Reward" on p. 246 Lucius says to Dumbledore "you're come back." It should say "you've."