Continuity mistake: When Gadget is driven to the TARDIS so the Doctor can retrieve it, the landscape around it is much flatter and without the large boulders strewn around the TARDIS when the Doctor arrived on Mars at the beginning. (00:52:41)
Continuity mistake: When the Doctor leans on the railing above the glacier while telling Adelaide about the Ice Warriors, he rests his left hand a ways out on the railing, in a spot where it should be visible but isn't in the next shot from behind him. (00:28:57)
Continuity mistake: When Martha arrives at Osterhagen Station 1, she unbuckles the belt of her teleport backpack, but in the wide shot immediately afterwards the belt is still buckled. In the next shot, it's back to being undone as she takes it off. (00:18:47)
Continuity mistake: When the Doctor, Rose and Jack get out of the TARDIS onboard the crucible, the console room is lit in very blue light. But in the shots of the console room before and after, it's lit completely differently. (00:09:15)
Continuity mistake: When the TARDIS is breaking through the time limit to reach the stolen planets, the console room catches on fire from the strain. But after reaching its destination, the console room is abruptly back to normal. (00:35:05)
Revealing mistake: When the Doctor and Donna run back inside the TARDIS to find the console room lit up in red and the Cloister Bell ringing, what's visible outside of the doors as they run in isn't the bright Shan Shen street the TARDIS is parked on but the dark soundstage the set was built in. (00:47:49)
Continuity mistake: After the TV signal cuts out due to the crash of the Titanic, when Sylvia says "It's gone dead" while searching for a channel, she moves her arm with the remote forward the same as she did in the previous, wider shot. (00:18:57)
Factual error: Kandoka labour laws require companies like Kerblam! to have a minimum of 10% organic staff. The villain claims that, as a result, only 10% of humans are employed, at all. These are two completely different quantities: if 10% of the cars in one parking lot are blue, this doesn't indicate that 10% of all cars are painted blue, but that's effectively what the episode thinks. The fact that no-one else, including the Doctor, calls him on this further magnifies the mathematical error.
The Lazarus Experiment - S3-E6
Continuity mistake: When the Doctor is saying goodbye to Martha, he's leaning on the open TARDIS door with his hand open. When Martha says that things just kind of escalated, the Doctor's hand is suddenly a fist. (00:02:00)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When Frank is dragged off by the pig-slaves, the manhole opening into the theatre is depicted as completely open from below. When the shot switches above to the stockroom when Solomon pushes the Doctor away from the manhole, the lid is resting half over it, before it being completely open when Solomon moves to pull the lid over the manhole and close it. (00:23:14)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When the Doctor turns the newspaper towards Martha so she can read the headline, in the wide shot, the corner in his right hand is bent back towards him. In the close-up showing the headline, the corner is straightened with his hand in a different position. (00:04:29)
Continuity mistake: When Martha, Milo and Cheen are talking in the powered-down car, the lighting changes between wide shots and individual close-ups.
Continuity mistake: Martha reaches her right hand up to open the window, prompting Julia to panic and say "Don't! We'll lose all the air!" When the shot changes as Martha points out that the windows aren't exactly airtight, her right hand is suddenly lowered. (00:09:47)
The Christmas Invasion - S2-E2
Continuity mistake: After the Doctor's challenged the Sycorax leader to a swordfight, when he's asking about the rules of combat while taking his dressing gown (bathrobe) off, he starts taking it off beginning with his left arm in two successive shots. (00:43:58)
Continuity mistake: When the Connollys are watching TV, in the shots facing the chair and sofa, the chair Eddie was sitting in is further away from the couch than in the shots from above, where the chair is almost touching the couch. (00:04:46)
Continuity mistake: Jackie gets Elton to take his shirt off by deliberately making a huge red wine stain on the front. When Elton comes back to the estate with the pizza, his shirt is unstained. (00:27:17)
The Christmas Invasion - S2-E2
Character mistake: When Llewellyn is describing what was carried onboard Guinevere One, the UNIT computer displays the space probe's name misspelled "Guinivere" One. (00:26:50)
Continuity mistake: When Indra Ganesh offers Joseph Green the briefcase with the Emergency Protocols, he's holding it with the handle in his right hand in wide shots, but with it in his left hand in close-ups of him. (00:12:36)
Visible crew/equipment: During the news broadcast on the disappearance of the Prime Minister, shown on an in-universe TV screen, the reflection of the camera filming the TV is visible. (00:12:00)
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)
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.