When the TARDIS returns to Christmas, The Doctor has to tell one of the children that it is his spaceship, but in the very next scene when Clara is looking at the children's artwork, one clearly shows the TARDIS.
When The Doctor is speaking to Clara before landing on the Cyberman ship, he tosses the telephone receiver (from the front door) on the console. When she calls again, the phone is once again on the hook in its door cubby and he has to retrieve it while being shot at by Cybermen.
When The Doctor and Clara are roasting marshmallows, just before The Doctor tells her that the pink ones are the best, Clara's marshmallow can be seen to be darkly toasted. After it cuts to The Doctor and then back to her, the marshmallow is suddenly raw.
When The Doctor and Clara are visiting the Papal Mainframe ship, it is said that their clothes are merely a holographic illusion projected over their bodies (to meet the Mainframe's nudity requirements while maintaining Clara's sense of modesty). Clara is very clearly wearing a bra. While the hologram could project the appearance of a bra over he skin, it would not offer the support that a real bra does. Clara is also wearing high heeled shoes. The Doctor was barefoot before creating the holographic clothing, suggesting that not even footwear is allowed on the Mainframe's ship, so her shoes would be illusions as well and not capable of raising her up several inches the way her high heels do.
In several shots when the old Doctor is talking to Clara, his hands are missing the aged makeup.
When The Doctor first arrives at Clara's estate, she looks out the window to see the TARDIS arriving and it is seen through he kitchen blinds at a very low angle, suggesting she is only a couple of floors above ground level. She is then seen descending many sets of stairs to reach to ground level. The angle she viewed the TARDIS from should not have been possible from a window that high up.
When a man comes into the tower to tell The Doctor about the arrival of the Daleks, he replies saying: "It's okay Barnable, don't worry. I have got a plan". However, when the man leaves, The Doctor secretly tells Clara: "I haven't got a plan, but people love it when I say that". Since he was still inside the "truth field" that covered the entire town, he shouldn't be capable of lying at all.
The plot makes a point that The Doctor is out of regenerations, counting the War Doctor and the Tenth Doctor's regeneration into himself in The Stolen Earth (2008)/Journey's End (2008). However, in The Impossible Astronaut (2011), it was shown that The Doctor was killed for real since he was killed during his regeneration. Of course, it was revealed that that wasn't really The Doctor in The Wedding of River Song (2011), but it still suggested that he had another regeneration left.
Before The Doctor and Clara meet the couple walking down the street in Christmas, The Doctor tells Clara their cover story that they're a couple from the next town over. Technically, he was lying when he said that, which should not have been possible due to the truth field.
When the Silence appears, Clara does not recognize it, however, as a primary school teacher, she would have watched the video of the moon landing in 1969 many times. Day of the Moon (2011) showed The Doctor embedding the video of the Silence telling humanity to kill them on sight in that video, which Clara should have remembered.
While Clara is following The Doctor and Tasha to her chambers, at one point Clara puts her hands in her pockets. The "clothing" she is wearing is a holographic illusion projected into the cerebral cortex of any observers. She is not supposed to have any actual pockets into which to put her hands.