Revealing mistake: Despite having three suns, everything on Desolation only casts one shadow.
Character mistake: The Doctor comments that the destruction of Earth will cost 5 billion lives. At the time of the episode (2006, one year on from when Rose met the Doctor), the population of Earth was around 6.5 billion. (00:25:10)
The End of Time: Part Two - S4-E18
Continuity mistake: When Wilf looks down while holding his service revolver at the end of his conversation with the Woman in White, the position of his right index finger changes between shots. (00:23:35)
Silence in the Library (1) - S4-E8
Continuity mistake: When the girl recognizes the Doctor on her TV, telling him that he was in her library, and he reacts, the position of her pencil crayon in her hand changes between shots. (00:20:45)
The Pyramid at the End of the World - S10-E8
Revealing mistake: The Monks show the Doctor and company simulations of a future, lifeless Earth - but some visible trees reveal the origins of the footage as being from the aftermath of natural disasters. (00:24:10)
Continuity mistake: When Donna is sneaking around the Sontaran ship, she's carrying Martha's old phone, which she's using to stay in touch with the Doctor, and the rubber mallet from the TARDIS. There are times when one of the two objects, the phone more often, is missing from her hands with no explanation.
Continuity mistake: During Donna and the Doctor's conversation with an Ood at the reception who she approaches to ask if he's all right with being a servant, in shots facing the Ood, he's holding his translator ball higher than he is in wide shots and shots facing Donna. (00:11:50)
Evolution of the Daleks (2) - S3-E5
Other mistake: After the hybrid Dalek Sec explains the Daleks' plan to harness the power of a solar flare to the Doctor, when the Doctor asks how many minutes until the flare arrives, Sec says it's in 11 minutes. A fair amount of action and backstabbing later, when the Doctor and Laszlo arrive at the top floor of the Empire State Building and after the Daleks take back control of the elevator, the Doctor says that the solar flare arrives in 6 minutes. It is a considerable stretch for everything that happens between those two moments to take place in only five minutes.
The Impossible Planet (1) - S2-E11
Continuity mistake: After Ida retracts the roof to show the black hole, there is a shot where the Doctor and Rose are looking at it, with nothing above them, as he explains about it. However, other shots of the base's control room show that there are things in the room, such as wires connecting the central computer table to the walls, that should have been visible. (00:08:00)
Rise of the Cybermen (1) - S2-E8
Other mistake: Alternate Jackie is said to be turning 40. According to a newspaper Mickey finds early in the episode, the parallel universe is in the same year as the main universe. When Jackie's age (she claims to be only turning 39) is announced at the party, Rose does not in any way react as though that's different from the age of main-universe Jackie. The problem is that, given that Rose was 19 when she met the Doctor, and it has been, for main Jackie and Mickey, likely at least two years since the events of "Rose" (remember Rose's one-year accidental disappearance), this means Jackie would have had to have gotten married, and had Rose, before she turned twenty, and been a widow by the time she was twenty. Given that Jackie is from the UK, a first-world country, and that she seems to be in her twenties in "Father's Day", this is highly unlikely.
Suggested correction: "Highly unlikely" is not impossible, there are many cases of this sort of thing happening even in the age of contraceptives. The episode is also set during the 80's, so it's possible that Pete and Jackie were misinformed. There are other TV shows such as Raising Hope or Bump that use this as a major plot point - hell, the very next episode after Fathers Day features an approximately 12-year-old mother in the form of Nancy, the 17-year old mother of antagonist Jamie.
Character mistake: When the Doctor and Rose are walking back to the TARDIS at the end, he explains to her that Queen Victoria was hemophiliac. She wasn't. She was a carrier of the disease due to a mutation in one of her parents' reproductive cells (probably her father, Edward, Duke of Kent, because he was in his fifties when Victoria was born), but since she only had it in one of her X chromosomes, she didn't actually suffer from it as the gene for it is recessive, which is why most hemophiliacs are male. It is true that several of her children inherited the gene, and one of her sons was hemophiliac, but not Victoria herself. (00:40:55)
Audio problem: When Rose is sitting in the observation deck, after she looks at the tree cutting and suggests they might be related, when she puts it back down, it makes a "clink" sound, as if made of metal, despite being in a sort of cardboard cup. (00:15:20)
Continuity mistake: During Mickey's struggle with the plastic trash bin, the angles of sunlight and shadows keep changing between shots. (00:22:20 - 00:23:20)
Revealing mistake: When the Doctor fires a champagne cork into Auton-Mickey's head, the lack of spraying or leaking wine, even when the Doctor is seen holding the open bottle horizontally immediately afterwards, makes it quite clear that the champagne bottle is actually empty. (00:25:23)
The Husbands of River Song - S9-E14
Revealing mistake: When Hydroflax's head is sitting on the section of the TARDIS console with the telepathic circuits, the console is obviously sagging inward, as if made of rubber.
Factual error: By halfway through the episode, the plot is alternating between following Clara and Jac in London, the Doctor in the fictional Central Asian country of "Turmezistan", and Kate in New Mexico. Despite the three locations being scattered all around the world, only in London is it depicted to be nighttime at any point in the episode. Central Asia and the southwestern USA are both shown to be daytime, despite the two locations being over halfway around the world from each other, and the events all taking place at roughly the same time.
The Day of the Doctor - S7-E16
Revealing mistake: When Osgood tells Kate that they're using Derren Brown as the explanation for flying the TARDIS into Trafalgar Square by helicopter, the two women are standing in front of the TARDIS. Its door is slightly open, revealing an interior that is obviously a picture. (00:05:00)
Revealing mistake: Mels is shot in the stomach by Hitler, leading to her regeneration. Rory, when checking the wound, says he's going to try and stop the bleeding. However, when she stands up while glowing with regeneration energy, and in every scene afterwards where Mels/River is wearing that dress, there is no bullet hole or bloodstain visible.
The Day of the Doctor - S7-E16
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.
Silence in the Library (1) - S4-E8
Revealing mistake: Also applying to the second episode of the two-parter, "Forest of the Dead": The keyboards on the Library's computer terminals are very obviously standard Apple keyboards - this in an episode set in the 52nd century.
