After the shootout in the subway, Max has his glasses on. Then when Max is shown sitting across from Vincent they're off and he puts them on. Then when they show him get off the subway with Annie, he doesn't have glasses on again.
In the subway scene, Max and Annie run to escape Vincent but are really running toward him (the seats face forward and they run back past them). Meanwhile, Vincent is running forward and runs through at least three cars (position shown when the train makes a station stop in addition to progress being shown passing through doors while the train is moving). Max and Annie are one or two cars in front of Vincent yet the shootout takes place in the second car from the rear.
After Max and Vincent leave the Fever Nightclub, Max crashes his taxi into two parked cars. In the first shot of the front of the car after the crash the taxi appears to have no damage at all, the following shot the left front light is broken and there's damage to the side of the taxi. In the next shot, the damage is gone again.
Right before Max crashes the car, they pass by Staples Center in Los Angeles and it is green and red (they change it to green and red during the holiday season). Then when you see Staples Center throughout the rest of the movie (which occurs on the same night) it is back to the original colors of blue and red.
At the catwalk scene, right after Max tosses out the briefcase, Vincent is holding his gun with his right hand, and passes it to his left hand. At the next shot, he's holding the gun with his right hand again.
When they enter the elevator in the hospital, and Vincent asks what floor, in one shot he is pressing the buttons with his right hand, but when they show the buttons pressed, it is his left hand (as the thumb is pointing down).
While driving on a near-deserted road (going to Fever) what appears to be the flashing red and blue lights of a police car (presumably escorting the production vehicles) can be seen reflected in the cab's window.
A shot at the end of the film showing Vincent from a distance sitting dead in the subway, you can see someone's shadow in the next car crouched down and moving between where the two sliding doors meet.
Vincent tells Max that damage to the taxi will be covered by his General Liability Umbrella policy. An Umbrella policy and a General Liability policy are two separate types of policy and collision coverage to a vehicle would not be covered under either.
During the night club scene where there is a frantic evacuation of the building, one of the Asian bodyguards can be seen standing up with people dancing around him instead of the chaotic 'evacuation' being portrayed.