When Dr. Fall and Heppelwhite fight over the bottle of alcohol and it smashes against the wall, in the long shot there is no liquid on the wall but in the close-ups there is.
When Charles brings Mr Bauman his breakfast, he places the tray across his lap in bed while Bauman is looking out of the window with binoculars. The scene then cuts to what Bauman is watching but when it comes back to Bauman the tray is gone. We return to Bauman's POV. He then decides that he is finished looking and puts the binoculars down and the tray is back on his lap.
In the opening scene of "The Little Black Bag", a calendar is shown on the wall and it gives the date as Thursday September 17, 2098. In reality, September 17, 2098 will be a Wednesday.