A New Day in the Old Town - S2-E1
Other mistake: The killer from another dimension shape shifts into the appearance of the victim, leaving the victim's body. But it transforms into Charlie's appearance and leaves behind its former female nurse shape, hiding Charlie's body to dispose of later. It would have needed to transform from the nurse shape, not out of nothing. The original nurse's body was not at that location.
Suggested correction: The shapeshifter transformed into Charlie using his dead body which it then disposes of at the end of the episode.
But when the shapeshifter transforms into Charlie, he shouldn't be leaving the body of the nurse behind since he had killed the nurse before entering the hospital and transformed there into her. The only dead body in that last scene should be Charlie's.
Other mistake: When Peter is watching Richard Steig in the interrogation room and gets up to go in there himself, he tells Walter to stay there. On the live video screens behind him, you can see Peter enter the room before he actually does. (01:03:50)
A New Day in the Old Town - S2-E1
Other mistake: Just after Olivia wakes up in hospital, she's trying to explain what happened, and we hear a heart monitor getting faster and faster - the doctor looks concerned and says they need to get her pulse down. But when we cut to the monitor itself it shows her heart rate as 72 - a perfectly normal value.
Other mistake: In the scene where Peter and Lincoln are driving to question Daniel Hicks, a crew member is playing a practical joke that was somehow missed in editing. If you look at the left side of Lincoln's face, someone is hitting him with a laser pointer. This is definitely not part of the script (i.e. Sniper rifle scope), as no one is actively targeting members of Fringe in this part of the show. (00:16:20)
Other mistake: Peter and Dunham are alone in an apartment with blood coming in under the door. They are called and leave for the subway. Peter should have had to walk through blood to leave.