Deliberate mistake: Looking closely at the floor of Eleven's console room reveals that the part of the floor around the console facing the door has been discreetly raised up, which it isn't in any other appearance of this console room, so that Clara can ride her motorcyle inside it without going down two steps. This wasn't in any previous episode and is gone in the next one.
Other mistake: Billie Piper is listed as playing "Rose" in the closing credits. Rose was certainly mentioned, yes, but Piper in this episode was actually sentient superweapon the Moment, who merely took Rose's form, and was in all other respects nothing like the former companion.
Continuity mistake: Among the various contents of the Black Archive are visible several objects that shouldn't be there, because it is impossible for U.N.I.T. to have acquired them. In particular: A detonation pack from "Planet of the Ood", set in the 42nd century, a few thousand years in the future. The Supreme Dalek's shell from "The Stolen Earth"/"Journey's End" - it was shot and killed by Jack Harkness, destroying the shell. The restraint chair from "The End of Time", last seen on the Vinvocci's ship. The makeshift sonic screwdriver built by the alternate Amy from "The Girl Who Waited", since that version of Amy was wiped from existence. A different explosive carried by Gregor Van Baalen in "Journey to the Centre of the TARDIS", which is both set in the future and involves a complicated situation in which the events ended up technically not happening.
Continuity mistake: When the Doctor teleports onto the Dalek ship and announces himself, he's holding up a broken Dalek eyestalk in the wide shot as he throws off his hood, but then in the close-up he throws the hood off again, with his arm lowered. Then he raises his arm again, with his arm bent at the elbow, only for a wide shot to have his arm sticking straight up, with his elbow straight. (00:00:55)
Continuity mistake: After the Doctor yells at Handles for putting him on the Dalek ship, the Doctor grabs him and carries the head with his right arm, then switches the head to his left arm. However, in the close-up as the Doctor asks, "What does this question mean?" we see Handles is back on the Doctor's right arm, but next shot as the phone rings, Handles is carried on the Doctor's left arm again. (00:01:56)
Other mistake: While looking around the Doctor's room, one of the children's drawings Clara passes by has the TARDIS in it... when the TARDIS has been absent for all of the 300 years the Doctor has spent on Trenzalore up to this point, so none of the pictures existing now should have the time machine present. And it's not a particularly weathered picture, either, suggesting it can't be from the beginning of his stay on the planet. (00:31:47)
Continuity mistake: After Strax hits Clara with a newspaper she comes downstairs and he asks her if she needs anything. She asks for water and he places the bucket of mop water on the table in front of Clara. They have a conversation about the water and at the end of the conversation Clara looks at her newspaper and the bucket has disappeared from the table in front of her. Neither of them moved it. (00:26:45)
Continuity mistake: In the restaurant basement, while the Doctor and Clara are looking around while speculating on why the clockwork droids are harvesting organs, Clara moves a chain out of her way with her left hand before releasing it, only to be holding the chain again in the next shot. (00:44:10)
Continuity mistake: When Robin Hood has finished introducing the merry men to the Doctor and Clara, Clara says "You really are Robin Hood and his merry men" and she is shown with her arms parted away from each other. In the next shot, her arms are suddenly closer to her. Then in the next shot, her hands are now closed together. (00:08:25)