Revealing mistake: When the Doctor calls Rose when she is with the Daleks, you can see on the screen of her phone that the call has been going on for at least 57 seconds, but she just answered the call. (00:06:50)
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: When the Doctor and Rose are talking at the bay, the Doctor's hair is blowing in the wind, but it's not when he is shown in the TARDIS. Considering the image of him on the beach is a projection of him in the TARDIS, the two versions should match. The actor was obviously on the beach recording his lines with Rose and back on a set doing the inside scenes.
Revealing mistake: When Lilith bespells Shakespeare to write the end of Love's Labour's Won the way the Carrionites want it, there are several shots where a white fleck of some kind is visible in the lower centre of the screen, stuck to the camera lens. (00:17:10)
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)
Evolution of the Daleks (2) - S3-E5
Revealing mistake: When Dalek Sec picks up the remains of the radio and turns it in his hands, the "Hamilton" label on the front of the radio is written backwards, indicating that the shot has been flipped horizontally. (00:05:00)
Evolution of the Daleks (2) - S3-E5
Revealing mistake: When the people in Hooverville are meeting around the campfire, it's raining. However, everyone's clothes and hats are completely dry. (00:28:05 - 00:28:45)
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: In the close up shot of the boys' machine guns firing, in the background a white truck can be seen driving across the green countryside; not appropriate for the time period of the story. (00:25:25)
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 Family of Blood (2) - S3-E9
Revealing mistake: The pyrotechnic mortars used in the destruction of the Family's ship are visible in the shots leading up to their detonation. As the Family run from their ship through the field, the mortars are visible against the branches behind them. Shortly afterwards, they detonate, creating the illusion that the invisible craft has exploded.
Revealing mistake: When the Doctor goes to toss the trio of Christmas balls up, the two times after the first toss are re-used shots of the first toss. His movements of his right arm to bring up the controller are too exact to be coincidence. (00:50:15)
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)
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.
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: When the Doctor and Donna are walking through the underground tunnels towards Mount Vesuvius, tunnels that can be presumed to be fairly warm due to their connections to volcanic activity, their breath is visible - something that only happens in cold temperatures. (00:31:20)
Revealing mistake: Soon after the Doctor and Donna are blasted back there is fake snow on the camera, indicating there's a camera there. (00:34:15)
The Sontaran Stratagem (1) - S4-E4
Revealing mistake: In the beginning, the ATMOS system takes the car to the Thames, then runs it into the river. At first, the car runs forward normally. After the angle change, there's a huge white cloud of mist behind the car that dissipates quickly, meaning an air cannon of a sort was used to propel the car into the river. (00:02:35)
Silence in the Library (1) - S4-E8
Revealing mistake: When the Doctor notices the unnatural triangular shadow signalling the presence of the Vashta Nerada and points it out to Donna, the shadow appears over Donna when she's standing in front of it, showing that it's a special effect added to the shot afterwards. (00:11:10)
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.
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.