Plot hole: Near the end of the episode, the part-human Doctor is said to have all of the memories of the original Doctor. If this is true, then the part-human Doctor should remember the time the original Doctor spent with Rose, so shouldn't commit genocide or need Rose to help him to not be a man of killing.
Other mistake: When Miss Hartigan is testing the earpod control, the men start off facing her. She instructs them to turn to the right. They turn 90 degrees clockwise. She then tells them to turn to the left. They turn 180 degrees anticlockwise. However, they should have only turned 90 degrees anticlockwise, and end up facing her again. What they actually do is turn on the spot, as opposed to turning left. It takes a further instruction from Miss Hartigan for them to face her again. (00:24:57)
Continuity mistake: At the end of the episode, Donna piles suitcases in the Doctor's arms in preparation to leave with him in the TARDIS, and places a striped hat box on top. The bags stay that way for a while as the camera cuts back and forth between them, and you can see the stack of luggage as the camera films the Doctor from behind. Then, as soon as the camera cuts back to the Doctor's face when Donna leaves to give her mother the car keys, suddenly he has one bag under his arm and another in his hand, and the rest of the bags are on the ground with the hat box stacked neatly on top. In addition, when the scene cuts from Donna closing the hatch of her car to her and the Doctor standing by the TARDIS, a large floral-patterned suitcase has appeared, which she was not shown removing from the vehicle. (00:43:15)
Deliberate mistake: When the Doctor visits all his recent companions before he regenerates, he goes to a book signing of A Journal of Impossible Things, written by Joan Redfern's great-granddaughter. When she is signing the book for the Doctor, you can hear the scratching effect from the pen but you can see that the pen isn't touching the paper. (00:59:50)
Other mistake: The security guards shoot away the lock on the door, which they then kick down with no real effort. But the door would still be held in place by the hinges - they don't kick it with enough force to rip the hinges out, and even if they had, we can see that there's no damage on the hinge side of the door anyway to demonstrate their forcible removal. (00:24:20)
Continuity mistake: After Miss Evangelista is killed, we see her communicator as her "ghost" slowly fades. In one shot we see from a slight distance the lights left lit tick down to 3, then a short while passes, we cut back and the lights are back to 4. They then tick down to 3 again.
Plot hole: Davros said that the reality bomb would destroy all parallel universes. He fails in the Doctor's universe, but the Doctor said in an earlier episode that every decision creates a new parallel universe, like if Donna had walked out of the TARDIS with Jack, Rose, and the Doctor. She wouldn't have imbued herself with Time Lordness, so Davros would have succeeded. This means that all the universes, including the Doctor's one, should have been destroyed, because if the Reality bomb only succeeds in one universe, it would destroy all the others as well.
Continuity mistake: Near the end of "Silence in the Library," 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.