Continuity mistake: Near the end of "Silence in the Libary," a sweeping shot shows that they're standing between two freestanding bookshelves. But at the beginning of this one, when they're supposed to be in the same place, they're able to break through a wall into a separate room.
Continuity mistake: When the Doctor realises that the Vashta Nerada infesting the Library came from the books, he rubs the back of his head with his right hand, with his left hand in his trouser pocket. When Other Dave speaks, and the Doctor turns to him and discovers he's been eaten, he takes his left hand from the back of his head, with his right hand in his pocket. In the shot after that, his left hand is back in his pocket with his right hand in position as if he'd removed it from the back of his head. (00:18:30)skater49th
Continuity mistake: After the Doctor tints Anita's visor, as he explains to River that he did it to try and trick the Vashta Nerada into thinking they're already inside Anita's suit, the position of his right hand, holding the sonic screwdriver, and the fingers of that hand change position between shots. (00:10:25)skater49th