When the Duke is shooting at the handcuffed President, we hear the sound of very loud, echoing gunfire even though he is conspicuously using Snake's submachine gun equipped with a rather large silencer.
When Snake triggers the tracer after the fight in Madison Square Garden, Hauk follows the signal to the World Trade Center. But Snake did not take the tracer bracelet with him, he left it on the man at the Garden. It is not on his wrist for the rest of the film.
After Snake takes off in the glider, Charles Cyphers is seen in the control room wearing a shirt and tie. In the next shot, he is wearing a suit jacket, which he then takes off. He wears the shirt and tie for the remainder of the scene.
Air Force One is shown in flight as an aircraft with engines mounted at the back either side of the tail. The wreckage on the ground and the computer image on the USPF computer as it crashes are of a 737 type aircraft with the engines mounted on the wings.
When the USPF initially go into New York immediately after Air Force One crashes, helicopter 3 is shown landing, and a large chunk of the sticky back black plastic used to create its USPF livery is peeling off and missing from the tail section, revealing the green US Army livery underneath. (The UH1 Huey used in this scene was on loan from the Missouri National Guard.) When it takes off again a few moments later, the tail is back to perfect black.
When Snake first receives a signal on the President's Life Signs transponder, he radios Hauk then checks his mission timer. The timer reads 18:50:18 (and counts down to :14 seconds). As the scene is fading (at the :14 sec. mark), the timer changes to 17:50:14
When Snake drives the station wagon through the gauntlet of convicts on Broadway, they pelt the car thoroughly with broken bricks. Immediately afterwards we see the vehicle from several angles, filthy but not actually dented.
In the scene immediately after Snake departs from the World Trade building, and is walking toward the wreckage of the President's jet and loading his gun, as soon as he passes a "campfire" (which is providing the low-level lighting for this shot) you will see Snake's face turning to the right, as he looks at a crew member (with a camouflage hat) taking his picture with a "handheld" camera.
While briefing Hauk about "David 14," Rehme says the jet is "Seven miles out and closing." Even at a leisurely 300mph (the cruising speed of the Convair 580 aircraft the Air Force One model was based on), the plane would have reached them in under a minute and a half. Yet they have plenty of time to get to the control center and ID the aircraft before it even crosses the river.
During the sequence where Snake is being chased by the Crazies (and is about to shoot an oval pattern in a wall so he can break through), his eye patch shifts enough to reveal a perfectly good eye. This is commonly regarded as a mistake, but it is not, Snake still has his left eye, but he wears the patch due to a paralyzed iris, making it extremely sensitive to light.
The American flag on the right-hand side of Air Force One's tail is not reversed. In reality it would have the stars in the upper right to represent a real flag moving forward. However, the flag could have been changed in the alternate history within the movie.
When the bombs in Snake's arteries are deactivated, he raises his right hand to his left wrist to display the time on his countdown clock. When we see a closeup of the clock as time runs out, Snake's right hand is not there. But when we switch back to the previous view, his right hand is still pressed against the clock.
Toward the end, the helicopter lands just outside the prison walls, and it's broad daylight. The rest of the film takes place at night, even though the times given indicate clearly that the helicopter landed only moments before the subsequent action.
Right before Snake's fight with the Duke at the end of the film, he watches his timer which says 00:01:32. Then after the President shoots the Duke and Snake gets the micro-bombs deactivated he waits for 10 seconds and watches his timer which reads 00:00.02. However the time elapsing between the two times he checks his timer is 00:02:08, but according to what we see on the timer in the film only 00:01:30 should have past.
Snake and the others return to the elevator on top of the World Trade Center (immediately following the loss of the jet-glider). Sticking out near the jam, you can see the fingers of a crew member pushing the right-side door closed.