Continuity mistake: During the sepia-toned explanation of the Doctor and Clara's initial investigation into Sweetville, when they pose as a married couple to infiltrate the place, are accepted, and the Doctor says "The missus and I couldn't be more chuffed, could we love?" in a bad Northern accent, Clara's fake smile goes from closed-mouth to open-mouth between shots. (00:19:57)
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)
Other mistake: When Mr. Thursday goes to see Vastra, to hire her to investigate his brother's death, he doesn't see her face (she's a Silurian "lizard woman from the dawn of time") until she lifts her veil to take a closer look at a photograph he brought her, upon which he faints. However, Vastra's veil is quite see-through, enough that her non-human appearance is obvious to the audience even before she lifts it. The same thing happens when she goes to visit the coroner later - although he doesn't faint.
Plot hole: At the beginning, when the Doctor, Clara, Angie and Artie get off the TARDIS, the two kids are absolutely convinced that they're on the Moon, despite the Doctor explaining that they're actually at the Spacey Zoomer ride on Hedgewick's World, the greatest amusement park in the universe. The kids only realise they really aren't on the Moon when Webley emerges from inside of a fake rock formation, followed by the soldiers. The part of the room that the soldiers come from, which should have been fully visible to the time travellers from the beginning - in fact, they should have been looking directly at it when they leaned out of the TARDIS in the very first shot - looks nothing like the Moon and contains a sign for the Spacey Zoomer ride and a large map of the amusement park. (00:00:01 - 00:00:45)
The Name of the Doctor - S7-E15
Continuity mistake: When the Doctor pulls the Intelligence's hand off his chin, Simeon's thumb is suddenly no longer touching his face. (00:25:56)
The Name of the Doctor - S7-E15
Continuity mistake: When the Doctor is talking to River just before entering his own timeline to rescue Clara, in the shots facing him, the glowing column representing his timeline is not visible, despite the fact that a wide shot reveals he's standing right in front of it. (00:39:25 - 00:40:04)
The Day of the Doctor - S7-E16
Continuity mistake: In this episode, I. M. Foreman's scrapyard is at the end of the alley next to Coal Hill School. In "An Unearthly Child", the scrapyard was far enough away from the school that Barbara and Ian take Ian's car to get there. Barbara also describes it to Ian as "there's a big wall on one side, houses on the other." If it was right next to the school, she would have mentioned "that scrapyard next to the school." (00:00:15)
Suggested correction: The evidence to this is merely a sign directing people to the scrapyard. This does not indicate whether or not the scrapyard in question is actually at the end of that road.
That still seems to be a point to bring up. Or better yet, "Her address is the same as the scrapyard on that sign outside the school."
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)
The Day of the Doctor - S7-E16
Continuity mistake: When the Eleventh Doctor takes off his fez, before throwing it into the time fissure, the position of his right hand on the hat changes between shots. (00:27:40)
The Day of the Doctor - S7-E16
Continuity mistake: When Clara asks Kate why her two assistants have entered the Black Archive, she rests her left arm on the sill of the window of the containment unit she's inside. In the next shot, when Clara slowly turns to look at "Kate", who's just revealed she's actually a Zygon imposter, her left arm is by her side. (00:39:55)
The Day of the Doctor - S7-E16
Continuity mistake: When the Doctors figure out how to disintegrate the door, when Eleven takes out his screwdriver to check the calculation and announces that it's complete, Ten's right hand, with which he's holding his screwdriver, goes from raised at roughly shoulder height to much lower between shots. (00:44:50)
The Day of the Doctor - S7-E16
Continuity mistake: After Clara enters the Doctors' cell, and they're talking about how the door was unlocked, shots inside the cell show the door about halfway closed, while a shot from outside looking into the cell shows the door wide open. (00:45:25)
The Day of the Doctor - S7-E16
Continuity mistake: When Osgood discovers Kate trussed up where the Zygons left her, when she pulls off the sheet covering her Kate's head is completely covered by a membrane-like mass. Moments later, after Osgood has begun working at Kate's hands to get her free, in the next shot with Kate's head in it her face is now uncovered. (00:46:15)
The Day of the Doctor - S7-E16
Continuity mistake: After the 10th Doctor marries Elizabeth, they kiss. While they are kissing, the collar on his jacket gets turned up, from her point of view. The camera then switches to look at her from over his shoulder, then back, and now the collar is straight. (00:49:40)
The Day of the Doctor - S7-E16
Continuity mistake: During the scene in the barn where the three Doctors have their hands on the button, when Eleven notices Clara crying and asks her what's wrong, the Doctors' hands go from stacked on top of each other on the button to only touching it with their fingers. (01:02:40)
The Day of the Doctor - S7-E16
Continuity mistake: When the three Doctors have all just come up with a plan to stop Gallifrey being destroyed, the War Doctor shouts "Thank you, Bad Wolf Girl!" At this moment, standing in front of "Rose", the Tenth Doctor's hair is spiked up and messy, but then immediately in the next shot, after the Moment has disappeared, when he questions what the War Doctor just said, his hair is flat and neat again. (01:06:25)
The Day of the Doctor - S7-E16
Continuity mistake: In the scene at the end in the Undergallery, when Ten walks up to the Gallifrey Falls painting, wondering what its actual title is, the door of his TARDIS, visible behind him, is open. In the next shot with the TARDISes visible, the door is shut. (01:10:55)
The Day of the Doctor - S7-E16
Continuity mistake: At the end, when the Tenth Doctor steps into his TARDIS, there are three sets of cups and saucers on the side of the bench farthest from the TARDIS, and Clara's set in the middle of the bench, but as the TARDIS fades away, there are two sets on each end of the bench. (01:14:10)
The Day of the Doctor - S7-E16
Continuity mistake: When the War Doctor kisses Clara goodbye, his tea cup rests on top of the saucer, but when he steps into his TARDIS the cup sits on the bench, beside the saucer.
The Day of the Doctor - S7-E16
Deliberate mistake: Looking closely at the floor of Eleven's console room reveals that the part of the floor around the console facing the door has been discreetly raised up, which it isn't in any other appearance of this console room, so that Clara can ride her motorcyle inside it without going down two steps. This wasn't in any previous episode and is gone in the next one.
Suggested correction: The Tardis has constantly shown it can change its internal structure to fit its needs or desires.
Suggested correction: Perhaps they didn't see everything and only focused on small things, or they remembered the Futurama episode with an amusement park set on the moon.