The initial American release of the book featured some extra lines in which Dumbledore goes into more detail about his plan to hide the Malfoy family from Voldemort. These lines do not appear in the British edition or later reprints of the American edition. It's thought that Rowling decided to pare down Dumbledore's explanation in a later draft, and the original American edition printed an earlier draft of the scene by mistake. The extra lines consist of Dumbledore explaining that they would make it look like the Order had murdered the Malfoys, which the Death Eaters would buy because that's what they would do.
Alfonso Cuarón, who directed the third installment before having to sit out of the fourth, would have loved to return, but he was not called.
Bob Hoskins was considered for Horace Slughorn, but he was rejected for Jim Broadbent. This could be considered the second time Hoskins had been denied a role in the franchise, as Rowling had previously promised to write a role for him in the seventh book, but she completely forgot about this and wasn't reminded until they shared an interview after the book was written.
Jamie Campbell Bower auditioned to play the teenage Riddle. He didn't get the part, but they cast him as the young Gellert Grindelwald for the next film.
Naomi Watts was approached to play Narcissa Malfoy, but her agent rejected it without her ever knowing it.
Emma Watson considered not returning due to worrying about not being able to keep up with her schoolwork. She ultimately decided she wouldn't be able to bear seeing someone else in the role. Hilariously, she then realized what a Hermione-ish thing she almost did.
A line in the script established Dumbledore as straight. Rowling nixed the line, writing on the script "He's gay" (this was well before Rowling outed Dumbledore in an interview). Writer Steve Kloves clarified in a discussion with Rowling (which can be seen on one of the DVDs) that he intended the relationship mentioned, about a woman that Dumbledore was fond of, to be platonic and that he suspected Dumbledore was gay, but removed it due to not wanting to cause confusion after Rowling made the notation.
The shooting script had Harry taking Dumbledore's wand after his death. Then Harry Potter and the Deathly Hallows came out, and the importance of Dumbledore's wand in the plot meant that that idea had to be scrapped.
All the Voldemort flashbacks from the book were in the original script. David Yates cut them down to Dumbledore meeting him at the orphanage and the two talks with Slughorn.
Rufus Scrimgeour was originally going to appear, but he was saved for the final film.
Concept art exists of Dumbledore’s funeral taking place on the island he was shown to have been buried at in the following installment, but his funeral was replaced with the wand-raising scene in the final film. Mad-Eye Moody is depicted as one of the attendees of the service, as he was in the book, but he too ended up cut from this film.
The appearance of the Carrow twins was supposed to tie into Draco Malfoy traveling between the Vanishing Cabinet in Hogwarts and the one in Borgin and Burkes, but this connection was not highlighted in the final cut of the film.
