
Continuity mistake: When the Doctor shows Clara the "listen" written on one of his blackboards, the handwriting is different to when it was first shown. (00:07:07)
Continuity mistake: After the Doctor is knocked unconscious, Clara uses the psychic circuitry to pilot the TARDIS to what turns out to be a moment in the Doctor's past, in the barn first seen in "The Day of the Doctor." The loft in the barn where the young First Doctor is sleeping in bed, however, was not present in the previous episode.
Suggested correction: These events take place thousands of years apart. Buildings (especially wooden ones) are likely to change in that time.
The barn appears three times, in "The Day of the Doctor", "Listen" and "Hell Bent." Its appearance in "The Day of the Doctor" is chronologically in-between its appearances in "Listen" and "Hell Bent." Both of the two latter episodes have the loft, but "The Day of the Doctor" doesn't. That's the mistake here.
Time Heist - S8-E5
Continuity mistake: After Delphox taps on the glass to awaken the Teller, her fingers close as her hand drops below shoulder level. After the angle change, her hand is above shoulder level, index finger extended. (00:06:03)
Time Heist - S8-E5
Continuity mistake: When the Doctor takes the case with the neophyte circuit out of a safety deposit box and opens it, his hands holding the case change position between shots, being suddenly raised higher in the second one. (00:31:27)

Time Heist - S8-E5
Continuity mistake: Near the end, when the Doctor and Clara are standing by the TARDIS on the planet they took the Teller and its mate to, Clara's hands go from touching the buttons on her jacket to loosely clasped lower down. In the third shot of her, they return to the first position. (00:44:28)
Continuity mistake: When Courtney comes into The Caretaker's room in search of paper towels, there is a "keep out" sign (reading "Go Away Humans") hanging in the window of the inner door. When the Doctor is talking to her just before she leaves, the sign is missing. Over the course of the rest of the episode, the sign proceeds to appear and disappear a few more times. (00:15:50)
Factual error: In this episode, it is stated that the Moon's sudden increase in mass caused a devastating global high tide. The amount of the increase in mass is given as 1.3 billion tonnes. But the Moon has a mass of over 70,000,000,000 billion tonnes, or over 50 billion times the supposed increase. Such a negligible increase in mass, less than 0.000000002%, would have almost no effect on Earth's tides, let alone be devastating. (00:20:46)
Factual error: When Clara is watching the Earth through binoculars in order to see whether people left their lights on or off, the planet is rotating at an impossibly high speed. (00:36:15)
Factual error: Near the end of the episode, the full moon is shown in the daytime sky, well above the horizon. This is impossible, as any celestial body lit by the Sun has its full phase only when it is directly opposite the Sun; thus, a full Moon rises at sunset and sits high in the sky only during the night. For the Moon to be in the sky during the day as shown, it would have to be visibly of a phase other than full.
Suggested correction: The moon is not actually completely full. It's in a gibbous phase (opposite of a crescent), which can indeed be seen in the daytime.
Visible crew/equipment: On the Moon's surface, the camera and boom mike can be seen reflected on the visor of the astronaut as she talks to Clara.
Mummy on the Orient Express - S8-E8
Continuity mistake: When the Doctor offers Professor Moorhouse some jelly babies in a cigarette case, the position of the candy in the case changes between shots. (00:14:10)
Mummy on the Orient Express - S8-E8
Continuity mistake: When Clara tells Maisie that "Difficult people can make you feel all sorts of things", as she finishes the sentence the view cuts from a close-up of her to a wide shot, and the light and shadows cast onto her face change with it. (00:17:20)
Mummy on the Orient Express - S8-E8
Continuity mistake: When the Doctor takes a phone handset from the wall and sonics it before calling Clara, his grip on the sonic screwdriver changes between the wide shot and the close-up. (00:21:45)

Mummy on the Orient Express - S8-E8
Continuity mistake: When the Doctor phones Clara and discovers she's trapped in the locked car with Maisie, his phone keeps switching hands. (00:21:55)
Factual error: After PC Forrest is killed by the Boneless, her nervous system is flattened and displayed on the wall, appearing as a mural. However, there are far too many nerves around where her liver would have been and not nearly enough around her brain.

In the Forest of the Night - S8-E10
Continuity mistake: When Clara mentions worrying about a witch in a gingerbread house while she and the Doctor are searching for Maebh, the phones she is holding switch hands between shots. (00:20:11)
In the Forest of the Night - S8-E10
Continuity mistake: When Clara is looking at the drawing in Maebh's homework book while the Doctor tells her about the solar flare, her grip on the book changes between shots. (00:22:44)
Dark Water - S8-E11
Factual error: When Clara is speaking on the phone to Danny at the start of the episode her phone shows the call waiting screen for an incoming call from him. (00:02:25)

Dark Water - S8-E11
Continuity mistake: When it's revealed that the dream patch is on Clara's hand, it's in her right palm. But in the wide shot when she's getting up as the Doctor explains the true nature of the patches, it's on her left hand instead. (00:13:21)

Dark Water - S8-E11
Continuity mistake: As Clara nods and says "Fair enough" when she thinks the Doctor has told her to leave, her arms go from crossed to by her sides between the closeup and the wide shot. (00:14:29)