Revealing mistake: When the Doctor and Donna are attempting to smash open a window while stuck in the window-washing cart, there is a shot of them from above just before Foster cuts the cable in which banging sounds are heard and the cart is shaking, but if you look closely at them they aren't actually touching the window at all. (00:26:10)
Revealing mistake: Max Capricorn is a cyborg, with his head being the only remaining part of his original body, wired up to a mobile life support system. If you look closely during his scenes, whenever he turns his head, the costume the actor is wearing to hide his body becomes apparent, as the wires and cables below Capricorn's head also move slightly.
The Family of Blood (2) - S3-E9
Revealing mistake: In the penultimate scene, with Tim Latimer helping an injured Hutchinson across a muddy battlefield at night, one of the shots is flipped - in the unflipped shots, Hutchinson has his right arm slung across Tim's shoulders, but in the flipped shot, it appears to be his left arm, and the two young men have switched sides. (00:39:45)
The Lazarus Experiment - S3-E6
Revealing mistake: When the Doctor and Martha get out of the elevator and enter the reception room in search of Lazarus, they pass by the string quartet. The violinist and cellist are only miming the actions of playing, as their bows aren't actually touching the strings of their instruments. (00:18:05)
Revealing mistake: After arriving in New New York's undercity, the Doctor and Martha find a screen playing the news. The Doctor tinkers with it and gets a video of flying cars passing a picturesque view of the city. This is a clip from the last episode set on this planet, "New Earth" - the fact that in a close-up of the screen, the TARDIS can be seen parked on the cliffs as it was in that episode clinches it. (00:03:55)
Revealing mistake: When the Gamestation's transmissions are cut off by solar flares, there is a close-up of a screen going over to static. In the upper right corner, a box with "Play" written in it is visible, revealing the image on the screen as a video. (00:34:50)
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)
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)
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: 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)
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)
A Good Man Goes to War - S6-E8
Revealing mistake: Several of the hallways in Demon's Run, particularly the one where Madame Kovarian has the baby brought to her and the one the Doctor's standing in when he realises where Amy and Rory's baby was conceived, are the repainted sets of the TARDIS hallways from "The Doctor's Wife", three episodes before.
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)
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)
The Christmas Invasion - S2-E2
Revealing mistake: When Daniel Llewellyn is looking up U.N.I.T. staff records on the computer, at one point a control panel for a DVD playing program is visible onscreen, showing that the special U.N.I.T. user interface is a video. (00:26:25)
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: In the Torchwood building, a large group of Cybermen start shooting at the Daleks and the Genesis Ark. The Daleks shoot several of them, and you can see a Cyberman that, thanks to hilarious CGI, just collapses like a sock without any physical regard that he is wearing tough metal leg armour, is floating a little off the ground, and falls way too fast than physical laws could allow. (00:25:15)
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.
Revealing mistake: At the end when the Doctor makes it snow, in the wide shot you can see the snow blows in from the left side, showing that they used a snow machine just off-camera. (00:54:30)
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.