Basketball - S1-E5
Corrected entry: In the opening shot, after Michael says "secret sign", a microphone appears at the top of the screen. (00:00:30)
The Alliance - S1-E4
Corrected entry: As the camera switches back and forth between Jim and Dwight as they discuss their plans to form an alliance, a microphone is seen hanging between the two actors.
Correction: The show is supposed to be a documentary, so any visible filming equipment can be considered proper in-universe.
Corrected entry: In the first episode, Michael and Dwight are in the conference room about to announce the possible downsizing. When Dwight says "I give you permission," you can see outside the conference room on the back wall near the map, a boom and camera man's shadow. (00:11:10)
Correction: You can see the camera crew on purpose. The camera crew there is not a mistake because they are documenting all the people in the office. That's why the characters are having confessions and talking to the camera all the time.
The Alliance - S1-E4
Corrected entry: After Roy catches Jim and Pam too close for his comfort, Jim reveals The Alliance joke. Then we see Dwight with the interview cam, while he has blonde hair. He had his normal hair just after the incident. There would be no reason for him to think that he needs to dye his hair at that point, since he had found out that Jim was messing with him.
Correction: Dwight had already applied the peroxide. It took time to bleach out.
Correction: After Jim explains himself, Dwight still gets his hair dyed, meaning he still believed that Jim was genuine with him.
Correction: The microphone being there was purposeful, the plot of the office is that it's a relatively low budget documentary series about an office. Any and all mistakes where equipment was seen were intentional to underline the possible inexperience of the crew in the TV shows universe.