Grifting 101 - S6-E9
Continuity mistake: During Professor DeSalvo and Jeff's argument, the amount of scotch in DeSalvo's glass changes a few times between shots. (00:08:40)
Basic Crisis Room Decorum - S6-E3
Continuity mistake: Chang is holding a piece of paper by the top, showing it to everybody. In the next shot, he is holding it by the side. (00:04:15)
Advanced Documentary Filmmaking - S4-E6
Continuity mistake: When the group is making posters for the MacGuffin Institute meeting, Annie holds up a poster saying "Welcome Archie" to show off name brand glitter but the character of Archie isn't introduced until the next episode (in which the banner shows up again). (00:03:11)
Suggested correction: This isn't a mistake. The episodes were simply aired out of order than the order they were produced. "Economics of Marine Biology" (production code 406), the following episode, was filmed before "Advanced Documentary Filmmaking" (production code 408).
Intermediate Documentary Filmmaking - S2-E16
Deliberate mistake: This episode is meant to be entirely a documentary made by Abed. At the end of the episode, there's a shot where the gang is looking into Pierce's window, except the camera is right in front of the window they're looking through. If Abed was filming this he would literally be right in front of the gang, completely obscuring their view through it.
Queer Studies & Advanced Waxing - S6-E4
Continuity mistake: At one point earlier in the episode, Elroy mentions that there are two baby birds in the nest that needs to be moved. Later, after the nest is moved, Abed mentions that there are three baby birds - two died and one survived.
Other mistake: Since Annie had completed the terrarium the group should have at least received a passing grade since the Professor made them a group of 7.