Factual error: When Mel Gibson is leaving NYC by car they come out of the Lincoln Tunnel onto route 4, that is impossible. The only way out of the Lincoln Tunnel is onto route 3. Then when they are driving through New Jersey on route 4, Mel Gibson says "There is the exit for 208." While you can get to 208 from route 4, they were driving on route 4 east, heading back towards NYC, not route 4 west which is the way onto route 208 they showed in the movie.
Factual error: Police officers, FBI agents, any law enforcement employees are not allowed to collect on rewards. It is their job to return kidnap victims, catch criminals, etc.
Factual error: When the FBI agents are in the apartment they notice a flash from some photographers outside the window and they rush to close the drapes. A flash is only good for about 20-30 ft. and any half decent paparazzi would be using a fast film and no flash so he wouldn't be noticed.
Answer: While the police would take fingerprints, it would take a long time to get all the fingerprints to the lab and tested for a match. They probably wouldn't have known whose prints they had at the time of the news broadcast. And even if they knew they had Gary Sinise's prints, they wouldn't automatically suspect him.