Plot hole: When Robin and Batgirl are defrosting the mirrors on the telescope, don't you think that at least some of the beams would reflect back at them?
Plot hole: After Freeze blasts Robin with his cold ray he just picks the diamond out of his hand with no problem. Why didn't the diamond freeze along with Robin?
Factual error: If everyone had been frozen for that length of time, their blood would crystallise, which would kill them. I imagine some of the buildings and anything made of glass might crack as well, but I don't really know about that.
Revealing mistake: In one scene where our heroes are defrosting Gotham city using the sunlight, we see Commissioner Gordon stepping from his car which has been covered in flexible icicles.
Continuity mistake: When Bruce and Julie go to the party in the observatory and are swamped by people, in the overhead shot at the start of the scene we can see that Gossip Gerty is trying to get to Bruce and Julie and that there is a man between Gerty and Julie. However, in the next shot (a tighter shot of the couple and onlookers), Gerty is suddenly right next to Julie, in the place where the man had been, with no time for him to move.
Deliberate mistake: During the opening battle scene, Mr. Freeze's gun gets stuck on top of a statue. Mr. Freeze then throws a chubby security guard up straight up into the air to knock his gun down. But it's completely wrong and defies the laws of physics. The security guard bumps the gun from the front, causing it to fall FORWARD and STRAIGHT DOWN, when in reality, the angle from which the guard smacked it in the front should have knocked the gun backwards even more. This reveals that the gun was just pulled down by a wire or was pushed forward from behind, as it completely defies the laws of physics and action/reaction.
Continuity mistake: In the last fight against Bane, Bane has got Robin and Batgirl by their throats against an icy wall. In a close-up of Batgirl, Batgirl's right hand is near Bane's spiked wrist guard. However, in the very next shot, her right hand is more near Bane's elbow.