Character mistake: Tina Fey fumbles her line and has to start it again. The original line should have been "How long do you think you're going to get away with this? Hazel or Jenna will know we're missing." Instead the second part of the line is "Henna or Jay; Hazel or Jenna will know we're missing!"
Continuity mistake: In the pilot episode, when Liz walks back into the writer's office Frank throws a sandwich at her, and she tells them to get it out of their system, i.e. throw things at her. We see all of the writers (Frank, Lutz, Toofer, Josh, a bald man, and a woman) grab various items off the table and the female writer grabs what appears to be an orange ball. When the shot changes to the objects hitting Liz, the ball does not appear. (It is not the bag of Cheetos, as you can tell it is made of some kind of foam.)
Continuity mistake: As Liz is talking to Dennis in the subway, he is leaning against one of the steel pillars with his left arm near to the location of his head. If you notice his arm is being supported by the pillar but his hand is not. In the shots of Liz whilst Dennis is leaning against the pillar, you cannot see his hand and the location of it should be covering part of his forehead.
Revealing mistake: In an early episode Jack introduces Tracey to all the TGS writers, but giving mini bios and facts about them in rapid succession. Seconds later we go to a flashback of Jack's assistant holding up pictures of random people as Jack names them. The problem though is you can see that it's the same picture being used every time.