Last of the Time Lords (3) - S3-E13
Factual error: When time is seen rewinding after the destruction of the Paradox Machine, the Earth and Moon are seen rotating and orbiting backwards. The Moon is depicted far closer to Earth than it actually is. (00:37:40)
Plot hole: At the beginning, when the Doctor, Clara, Angie and Artie get off the TARDIS, the two kids are absolutely convinced that they're on the Moon, despite the Doctor explaining that they're actually at the Spacey Zoomer ride on Hedgewick's World, the greatest amusement park in the universe. The kids only realise they really aren't on the Moon when Webley emerges from inside of a fake rock formation, followed by the soldiers. The part of the room that the soldiers come from, which should have been fully visible to the time travellers from the beginning - in fact, they should have been looking directly at it when they leaned out of the TARDIS in the very first shot - looks nothing like the Moon and contains a sign for the Spacey Zoomer ride and a large map of the amusement park. (00:00:01 - 00:00:45)
The End of Time: Part Two - S4-E18
Factual error: The Vinvocci ship is said to be orbiting 100,000 miles above Earth, later refined to 105,000 miles. That's one-third of the distance to the Moon, and the Earth's apparent size from the windows of the ship is far too large for them to be that distance, or even 100,000 kilometres above the Earth's surface. Judging by the Earth's apparent size, the spaceship is actually orbiting 6,000-7,000 kilometres above the planet's surface.
Plot hole: After getting out of the Pyrovile escape pod, the Doctor and Donna start running back to Pompeii. In the process, they are somehow able to outrun what appears to be a pyroclastic flow. Pyroclastic flows can reach speeds of up to 700 kph, and should be impossible for them to outrun. (00:38:15)
The Christmas Invasion - S2-E2
Other mistake: When Rose and Mickey are running across the plaza after being dropped off by a cab, a fence is visible between where they are and where the cab pulled up. It is unlikely to say the least that they would have gotten out of their cab right by a fence and climbed over it. (00:08:45)
Factual error: When the Royal Hope Hospital is transported to the Moon, the area it is in is dark, indicating they are on the side of the Moon not currently facing the Sun. However, in a wide shot of the hospital on the Moon, something that looks like the Sun is visible, something that the illumination of the landscape and the Earth shows cannot be visible. (00:07:40)
Utopia (1) - S3-E11
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. (00:35:15)
Utopia (1) - S3-E11
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." (00:34:40)
Factual error: When Clara is watching the Earth through binoculars in order to see whether people left their lights on or off, the planet is rotating at an impossibly high speed. (00:36:15)
Continuity mistake: After the Doctor and Wilf get off the seniors' bus, they're waving goodbye. In a close-up shot, the Doctor is waving with his right hand, but in a wider shot immediately afterwards, he's switched to his left hand. (00:25:10)
The Day of the Doctor - S7-E16
Plot hole: 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: In this episode, Rose's teeth look different than in all her other appearances. She also has a different accent, both for no apparent reason.
The Snowmen - S7-E7
Continuity mistake: As Clara first realises the differences in the TARDIS exterior and interior, she goes to circle the outside, leaving the door open. A second later, it's closed, but open upon her entrance.
Character mistake: The question to unlock one of the doors asks for the next number in the sequence 313, 331, 367. The Doctor gives the answer as 379 with a convoluted (albeit correct) explanation of happy primes, but there are much simpler solutions. The difference between the first and second numbers is 18 and the difference between the second and third is 36 (which is 2 x 18), suggesting that the difference between the third number and the next could be either 54 (= 3 x 18), giving 421 as the next number, or 72 (= 2 x 36), giving 439. It's likely that the Doctor would see that the answer is ambiguous, and yet he confidently states that the most complicated answer is correct. (00:08:45 - 00:09:15)
Suggested correction: If you know what a happy prime is, like the Doctor does, it might be simpler than your idea because you gave 2 options - 421 and 439. The idea of a pattern is there should be only one answer that fits, which would be the case if they're all happy primes.
Character mistake: After the Carrionites kill Lynley, the Master of the Revels, by using the doll to make him drown on dry land, the Doctor says, while examining the body, that he's never seen a death like it. In the Third Doctor serial "The Mind of Evil", involving a monster that could make people relive their worst fears, a character died in exactly that fashion. (00:13:10)
Factual error: The Olympic torch relay is presented as if there is only one runner carrying the torch through the entire city of London to the Olympic stadium to light the cauldron. Furthermore, when the torchbearer collapses, the TV announcer reacts as if it is now impossible to get the torch to the stadium, just before the Doctor steps in and picks it up. Any broadcast of any Olympic opening ceremonies would show that that's not how the torch relay has ever worked.
Suggested correction: Perhaps they didn't see everything and only focused on small things, or they remembered the Futurama episode with an amusement park set on the moon.