Plot hole: One of Snape's memories show Lily telling baby Harry to be safe, be strong. That had to have occurred before Voldemort killed Lily. Snape was not in the house until after Lily was killed and Voldemort was gone. Only Lily and Harry should have had that memory, but not Snape.
Plot hole: Immediately after the confrontation between Harry/McGonagall and Snape in the Great Hall, Harry runs up to Ravenclaw tower against crowds of students running the opposite direction. However, all the students would have been in the Great Hall during that confrontation, and therefore would have been coming from the same direction as Harry.
Other mistake: In both the books and the movies, Harry's Invisibility Cloak can always be seen until it is worn. In Diagon Alley, just before the Gringott's break-in, Ron pulls out the cloak to cover Harry and Griphook, but it turns invisible before he puts it on them. Ron merely mimes the motion of covering them.
Visible crew/equipment: During the main Harry and Voldemort duel in the courtyard, there is an aerial shot of the duel, with Voldemort on the left and Harry on the right. Between them, in one of the arches of the alley, a cameraman in black is visible for about one second.
Continuity mistake: When Harry arrives in Hogsmeade and meets Aberforth Dumbledore, his stubble disappears in one shot and then reappears later in the same scene.
Continuity mistake: Just after Harry and Lord Voldemort have their altercation in the Forbidden Forest, Neville is seen walking into Hogwarts' courtyard entrance to pick up the sorting hat. In the first shot, there is a camera angle of him from the waist down from behind, and he is stepping over the rubble and having no trouble walking. In the very next shot, you get a full frontal view of him walking, and he is limping and can barely move.
Continuity mistake: In Snape's flashback Harry's mother, as a child, has brown/black eyes. When she is an adult she has blue eyes.
Other mistake: In Snape's memories, Trelawny's prophecy is the wrong one. If you listen closely you can hear her saying "master shall be reunited once more". That's the prophecy that she told Harry in PoA, not the first prophecy Snape half-overheard Trelawny telling Dumbledore.
Continuity mistake: Throughout all the movies, the characters discuss several times how much Harry's eyes resemble his mother's. When we see flashbacks of Lily in this movie she has dark brown eyes. In past movies, when she was portrayed as a Hogwarts student, (in most of the same memories) she had blue eyes.
Visible crew/equipment: Right after Harry has seen Snape's memories in the pensieve, he turns around and sits down on the step and the camera can be seen reflected in the gold on the wall behind him.
Continuity mistake: When the students enter the Great Hall, Ginny is seen among them in full uniform. However, when one of the Patil sisters begins screaming, she can be seen in the casual clothing that she was wearing earlier in the Room of Requirement.
Continuity mistake: The pillow behind Griphook constantly moves between shots while the Trio is talking to him at Shell Cottage.
Revealing mistake: In the scene where Harry and co. are in the Room of Requirement, at several points during this scene, you can tell by the way the light shines that Harry's glasses are missing their lenses, especially in close up shots of Harry.
Other mistake: After Voldemort disapparates while Snape is dying, Harry asks Hermione for a flask for Snape's tears. She pulls a vial out of her purse. However, she hadn't had a purse during the entire battle, then has one at this moment. Afterwards, she no longer has the purse in any other scene.
Continuity mistake: During the final battle between Harry and Voldemort, Harry disarms Voldemort and his wand spins out of his hand counter-clockwise, then it cuts to the wand spinning towards Harry clockwise.