Continuity mistake: Near the end, when Jeff says to Troy "you're a man now" the camera goes onto Troy and we can see Abed asleep on Britta's shoulder (Britta is also asleep), even though in the shot before that, they were both awake. Then after the shot of Troy we see Britta and Abed awake again.
Character mistake: Annie when giving her gifts is holding two little mini flags to cover the possible ethnicity of Shirley's baby. One is a Chinese flag and the other is Zimbabwe. Annie makes reference to the fact that the ethnic food she made is one from Ghana. She makes Ghana ethnic food but does not get the right flag for Ghana. She decorates the table with numerous flags which are also Zimbabwe.
Revealing mistake: When Chang is getting blasted with the minigun, the safety pads under his shirt are very noticeable. (His shirt is extremely "puffed out" and you can see the outline of pads.) He's been tied up most of the episode and wasn't wearing pads earlier, so there's no way he was able to get pads on in the meantime.
Character mistake: At the party, Jeff is talking to Ted about his parents' divorce, which included a custody hearing. Jeff mentions the divorce lawyer drove a Mercedes SLK. However, the SLK didn't come out until 1996, which would make Jeff at least 21 (turning 22) at the time of the divorce (he was born Nov 1974). There would have been no need for a custody hearing (Jeff was an only child between his mom and dad) and his parents divorced when he was very young, so the divorce had to happen prior to 1996.
Character mistake: Right before Jeff searches Abed's bag, Britta explains to the group that Abed is the innocent one. After Jeff reminds her that Abed put gum in her hair, she looks at no-one in particular and says "empty the bag, Abed." Jeff is the only one holding the bag, with Abed standing five feet away from both of them. It's clear this line was directed to Jeff, but the wrong name was used.
Visible crew/equipment: When Jeff runs to the car which his father drives and opens the door of the driver, you can see the camera in the side window of the car. You can see it's a set camera and not a 'handheld type', the one that Abed is using during this episode.
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.