Continuity mistake: When the Doctor gets the rocket to work by tinkering with the boost reversal circuit, he picks it up with his left hand, sonics it, and then transfers it to his right hand, all in a shot where he has his back to the camera. In the next shot, facing him, he's holding it in his left hand as he pulls something off the device. The shot after that, facing Professor Yana again, the circuit is back in his right hand.
Character mistake: When the Futurekind spy starts sabotaging the silo's systems, and Chantho notes that they're losing power, she says, "We're losing power-tho!" - forgetting the "Chan-" prefix that she starts anything she says with, in order to not swear by her species' standards.
Other mistake: The stet radiation in the red room is able to disintegrate living beings, with the exception of the immortal Jack. People not going in the room are shown to stay out of the line of the door when it's being opened, lest they be disintegrated. When Jack comes out of the room after having finished in there, the Doctor, telephoning the rocket, is in line with the open door yet remains completely fine.
Other mistake: When Professor Yana shows Martha his pocket watch, which she comes to realise is a Chameleon Arch, she initially finds it familiar from the appearance of the back, and eventually asks him if she can see the front. Before she eventually sees the front of the watch, there is a shot where the front is hidden to the camera, but should be perfectly visible to Martha, just as Yana says "It's old. It's not meant to be."