Revealing mistake: When the landlord, Purcell, wakes up on his floor because his pet dog is licking his hand, we see a few pieces of dog food fall from his fingers - should have been a re-shoot, but all credit to the actor, he knows it has happened and is quick to hide them again. (00:38:38)
Revealing mistake: When the Cybermat gets free after the Doctor tries to hit it with a pot and accidentally gets Craig instead, it is very obviously CGI when it slides along the floor and stops by a piece of furniture. (00:26:34)
The Doctor, the Widow and the Wardrobe - S7-E1
Revealing mistake: When Cyril is walking around the statue and chair in the tower, if you look closely you can see the statue's hands move or shake. (00:30:14)
Dinosaurs on a Spaceship - S7-E3
Revealing mistake: The Doctor tells everyone to run, which they do. As they go around a corner in the distance, the actors at the rear of the group, presumably believing themselves out of shot, slow to a trot. (00:05:04)
Revealing mistake: When Jex's ship explodes, we see the smoke drifting over town. But it's really obvious that the smoke is coming from the next street, not miles away where Jex's ship is. (00:42:06)
Revealing mistake: When Brian is woken up by the cube he was watching suddenly moving, the cube is obviously not lying flat on the table surface, being slightly tilted up in order to make room for the device being used to make it spin. (00:16:41)
The Snowmen - S7-E7
Revealing mistake: When the Doctor and Vastra are on the TARDIS talking about Clara's injuries, and he says that he thinks for once the universe owes him one, he pulls some switches on the console. If you pay attention, you'll notice that the console wobbles slightly as he does so. (00:49:48)
The Bells of Saint John - S7-E8
Revealing mistake: During the sequence where Clara sets foot on the TARDIS for the first time, and she and the Doctor then stop the plane from crashing, there are some shots which make it fairly obvious that the mug of tea she's carrying is empty. (00:25:35)
Revealing mistake: While the Doctor and Clara visit the market in Akhaten, the rubber mask of one of the aliens they encounter is revealed, as the eyes of the actor wearing it are very much visible. (00:08:10)
Revealing mistake: Just before Strax leaves grumbling that he's going to play with his grenades, if you look closely you can see his face mask peeling. (00:27:45)
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)
Deep Breath - S8-E1
Revealing mistake: After the clockwork droids shut down, when the ones in the restaurant basement start falling over, the extended blade of one of the toppling droids flops about like rubber. (01:08:15)
Revealing mistake: When the old man is stabbed, the sword is retracted and we can see there is no blood visible on the sword. This is visible even before the blood is wiped off. (00:06:55)
Revealing mistake: When Robin Hood comes up behind the Doctor to push him into the water, it's obviously a stunt double with a bad wig.
Revealing mistake: During the scene at the beginning where people are taking pictures of themselves with the Cybermen, there are a few moments where the Cybermen's armour is revealed to be rubber when people touch it. (00:02:45)
Revealing mistake: During the sequence where the Cyber-clouds covering Earth are burned away, there is a shot showing the clouds burning over the Sydney Opera House. The shot is, very obviously, a still image to which CGI of the burning sky has been added, with extremely unconvincing results. (00:47:37)
Revealing mistake: In several episodes of season 9, starting at "Under the Lake" and continuing through "Before the Flood", "The Girl Who Died" and "Sleep No More", the same white hallway model is used. For the first two episodes, this makes sense, since they play in the same location. However, the same hallway is used in "The Girl Who Died", when the men from Ashildr's village get blasted to death, and again in several scenes in "Sleep No More" when the characters run through the hallways. Also, it seems the control room from the underwater research center, where the characters hide from the ghosts in "Under the Lake" and "Before the Flood", is reused as the white room in "Hell Bent" when the Doctor pulls Clara out of her timeline, though in "Hell Bent" the room is nearly empty and the room in "Under the Lake" and "Before the Flood" is full of equipment.
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.
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)
Revealing mistake: Despite having three suns, everything on Desolation only casts one shadow.
Chosen answer: 1) When Stephen Moffat took over he ignored a lot of what had been developed before (there is not in-universe answer). 2) It would have killed Rose, so the Doctor absorbed the energy. His body regenerated before the energy could do a significant amount of damage that would prevent regeneration.