Deliberate mistake: When Barry is on the phone with Kent for the 2nd time (right before Barry invites Kent to the station), we see Dan (Alec Baldwin) in the booth trying to get Barry to hang up on Kent. Dan holds up a sign that says "dump HIM." Problem is, we are looking at a reflection of the sign, so the sign was actually written backwards, as a mirror image.
Other mistake: Barry is supposedly dead at the end but when the screen brightens before the shot pans up, his head wiggles a bit.
Continuity mistake: Dan is talking to Barry about how he is the boss. In one shot, Barry is looking down with his eyes wide open. In the next immediate shot, which is wider, his eyes are closed.
Visible crew/equipment: At the end, the helicopter filming the shots of Dallas is blatantly visible in one of the buildings.