Continuity mistake: After the Doctor brings Amy onto the TARDIS, the material of the tweed jacket he's wearing is entirely different in the close-ups of the Doctor's hands, as he's fiddling with the gears on the panel. (00:59:45)
The Christmas Invasion - S2-E2
Continuity mistake: When Danny is killed he is reduced to a skeleton but when his 'corpse' is seen on the ground only a few bones and his skull are visible.This however is impossible as a full skeleton should be seen.
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.
Deliberate mistake: The young Amelia Pond (played by Caitlin Blackwood) has brown eyes, however her older self (played by Karen Gillan) has green eyes.
Audio problem: Just after the Doctor makes his appearance and Lynda tells him he's "in the house", she appears to continue talking to him but we never hear what she says; instead the scene cuts to the other two housemates complaining about the Doctor's sudden appearance. (00:01:40)
Factual error: The Globe has (and had) 20 sides, not 14.
Character mistake: David Tennant has a strong Scottish accent, and puts on the Doctor's English one. When David and Freema Agyeman are lying on the bed in the hotel, David forgets his English accent when he says, "There's something I'm missing, Martha."
Continuity mistake: When Yvonne Hartman is showing the Doctor through Torchwood after leaving the TARDIS, she goes to push a door open into the main building with one hand, but when the camera cuts to them entering the building she pushes them open with two hands. (00:20:10)
Plot hole: In this episode, the Doctor is said to have died during the events of "The Runaway Bride." As a result, he was not able to prevent the spaceship Titanic from crashing in London. It is later shown that southeast England was obliterated by the explosion. This falls far short of what the Doctor mentioned in "Voyage of the Damned", wherein he frantically repeated that if the Titanic crashed, the drive's explosion would wipe out all 6 billion people on the Earth (that's the entire human race at the time of the show's airing).
Last of the Time Lords (3) - S3-E13
Deliberate mistake: The Master uses his laser screwdriver to suspend the Doctor's ability to regenerate. While he is in fast motion, in a close-up you can briefly see Lucy's arms in fast motion as well. (00:12:20)
Continuity mistake: After the Doctor says, "Hang on, I missed it!" he smacks his head a couple of times and leaves a red mark on his forehead. When he turns his head immediately after that and says, "I saw," the red mark is no longer there. (00:32:35)
A Good Man Goes to War - S6-E8
Continuity mistake: Rory threatens Madame Kovarian with a sword. It's on her neck, then in the next shot the sword is not touching her. (00:23:38)
The Impossible Astronaut - S6-E2
Continuity mistake: When the Doctor is shot and killed while trying to regenerate on the beach, he falls with both legs extended. When River, Amy and Rory run up to him his left leg is bent. The camera cuts away again, and when it cuts back to show the Doctor's body one more time both legs are straight again. In addition, his left arm changes position between shots, between stretched out pointing away from his body on close-ups to bent with his hand resting on his thigh in wider shots. (00:08:15)
Plot hole: The Doctor leaves Martha a recording with instructions for her for when he's human. However, the flashbacks to the incident that led to him using the Chameleon Arch don't leave enough - or any - time for him to have made the recording, presenting his decision to use the device as a spur-of-the-moment one that he immediately acts upon.
The Time of Angels (1) - S5-E4
Continuity mistake: When the Doctor is ready to catch River Song, the TARDIS exterior shown is that of the Tenth Doctor's, not Eleven's. In particular, the St John's Ambulance logo is missing. (00:04:35)
The Impossible Astronaut - S6-E2
Visible crew/equipment: Just after Amy comes out of the bathroom, a boom mic can be seen popping up behind the agent's shoulder. (00:28:55)
Audio problem: While Billy Shipton is showing Sally Sparrow the TARDIS (in the room where he keeps the abandoned cars), just after he says, '[You] can't even get in it', he rattles the TARDIS' handle to demonstrate. Watch carefully: When he shakes the handle we hear a rattling noise as if the door is shaking, but in fact the door stays rock solid. (00:19:35)
The Impossible Planet (1) - S2-E11
Other mistake: When the Doctor asks what the planet is called, Ida says, 'Don't be stupid, it hasn't got a name, how could it have a name?' A few minutes later, she tells the Doctor the planet is called 'Krop Tor'.
Plot hole: Martha tells the kidnapping couple to switch off all their engines so the Macra don't notice them. Yet during this scene, the car remains hovering above ground, meaning the engines must be engaged in some way, and the Macra should still hear them. And these things don't hover when deactivated - the car was shown to be parked on the ground at the start of the episode.
Suggested correction: He would have been planning the change to human for a while now, not just made it up on the fly. So he could have made a recording before the opening scene.
The episode makes it pretty clear that he and Martha got taken by surprise by the Family's attack. The Doctor also explicitly says, when he gets out the Chameleon Arch headset, that he never thought he'd use it. The flashbacks consistently present his decision to become human as one made as an urgent, in the moment one.