Revealing mistake: When Britta slides through Jeff's legs to shoot the disco guys, you can see the skateboard-type thing the actress is lying on.
The Science of Illusion - S1-E20
Visible crew/equipment: After Britta accidentally bumps the gurney into the window, she grabs the body bag in a failed attempt to keep the corpse from falling out into the quad. Right as she grabs the body bag, look on the ground behind her. Someone slides a mat in behind her for her to fall onto. (00:13:50)
English as a Second Language - S1-E24
Continuity mistake: When the class leaves the Spanish classroom so Chang can talk to Jeff, in one shot Annie only leaves her audio recorder on her desk, but in the next shot her textbooks are suddenly there.
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.
Spanish 101 - S1-E2
Continuity mistake: When Pierce pours him and Jeff a glass of whiskey, two glasses are shown and he pours whiskey into only one of the glasses. In the next shot, both glasses have whiskey in them. (00:09:25)