Continuity mistake: When Rory finds his dad on the spaceship, Brian is lying on a gurney next to the two farthest slabs, which are empty. When Amy and the Doctor come through the portal to the spaceship, Brian's gurney is between the two closest slabs, with Rory lying on the closest one, which previously had a completely different person lying on it.
Continuity mistake: When the Doctor compares Amy's inaccurate drawing of Rory to the real deal and laughs, in a shot facing Rory the Doctor is shown lowering the picture, than in the next shot facing the Doctor he still has the picture raised. In addition, his grip on the paper changes from holding the bottom of the page to the side.
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.
Other mistake: Billie Piper is listed as playing "Rose" in the closing credits. Rose was certainly mentioned, yes, but Piper in this episode was actually sentient superweapon the Moment, who merely took Rose's form, and was in all other respects nothing like the former companion.
Plot hole: When the Doctor and Missy, the latest incarnation of the Master, meet for the first time, she passes herself off as an android. Only later in the episode, after seeing the Gallifreyan hard drive that she has been uploading human minds to, does the Doctor realise that she is a Time Lady, and he doesn't realise that she's the Master until she tells him in the episode's cliffhanger ending. The problem here is that over the show's long history, Time Lords have been shown to have the ability to recognize each other instantly, an ability that makes perfect sense given their powers of regeneration. Previously in the new series, it was even stated that Time Lords can psychically sense each other, an ability demonstrated on-screen in "The End of Time." In "Dark Water", this does not happen, at all. Now, the Master has a history of using technology to block his/her psychic signal before - the Archangel satellite network from Series 3. However, at no point in this episode or the next one, "Death in Heaven", does either the Doctor or the Master even mention such technology as being used, leaving the mystery of how the Doctor didn't recognize Missy as a Time Lady at the very least a rather glaring hole.
Factual error: The Koh-i-Noor, as depicted in this episode, looks absolutely nothing like the real diamond. The episode's gemstone is about the size of the palm of the Doctor's hand, and shaped in a stereotypical conical cut. The real Koh-i-Noor is much smaller and oval-shaped.
Continuity mistake: When the TARDIS crashes through the ceiling and lands in the tunnel, the side visible to the camera is not the one with the door. The camera cuts away, and when it returns to the TARDIS, on the same angle as before, the door is now facing the camera, as revealed by the sign and St. John's Ambulance medallion.
Continuity mistake: In the penultimate scene, just before the Doctor melts Amy's Flesh avatar, he points his sonic screwdriver, snapping it to its full length and opening up the four metal bits around the light at the tip. In the next shot, the screwdriver is "closed", but in the shot after that it's "open" again.