Continuity mistake: During Priori Incantatem, Voldemort's victims come out in reverse order. However, James comes out before Lily. He should come out last, because he was killed before Lily. (Corrected in later versions of the book).
Factual error: Harry says to Sirius in one of his letters that Dudley chucked his Playstation out the window. However, Book 4 takes place in 1994, and the PlayStation wasn't launched in Britain until 1995.
Continuity mistake: When Harry is tied up against Tom Riddle's tombstone, it says that his wand is at his feet. Later, Voldemort tells Wormtail to fetch Harry's wand so that they may duel. Wormtail then retrieves Harry's wand from over near Cedric's body.
Plot hole: In the Swedish hardback version. The trio is visiting Hagrid just after the revelation about the triwizard tournaments revelation. They are discussing it and Hagrid says "I never thought I'd see the tournament again, in my life" implying he'd seen it earlier. Although, this tournament is the first one in over 100 years. Hagrid is supposedly 65 years old, since he was expelled in his 3rd year 50 years ago (in CoS) making him 63 in CoS and 65 in GoF.
Continuity mistake: In chapter 22 of this book it says that Harry and Cho Chang are a year apart. But in Prisoner of Azkaban, it states they are two years apart.
Plot hole: In chapter 12 it says "The stranger sat down, shook his mane of dark gray hair out of his face, pulled a plate of sausages toward him, raised it to what was left of his nose, and sniffed it. He then took a small knife out of his pocket, speared a sausage on the end of it, and began to eat." But that should have tipped Dumbledore off that he wasn't the real Alastor Moody. The book states plainly that Moody never eats or drinks food he didn't prepare. This is a well known reason he was considered paranoid.
Continuity mistake: When Fred and George bet on the Quidditch match with Bagman, it says Fred put the paper with their wages in his robes, but Fred was wearing Muggle clothing.
Continuity mistake: When Dumbledore, Fudge, and Moody are talking about Crouch, at one point it says, "said Crouch angrily." Crouch wasn't in the room - they were only talking about him.
Continuity mistake: In chapter 35, the book says '"The Imperius Curse," Moody said.' However, the Polyjuice Potion has already worn off, so it should be Barty Crouch Jr. speaking.