Continuity mistake: When Ramius is shot, he drops to the floor holding his wound with his left hand. He is still using his left hand when Ryan asks "What happens if he makes it to that ignition circuitry". The angle then changes, and Ramius is holding his wound with his right hand as he answers "He can incinerate the ship". New angle change, and Ramius is once again using his left hand on the wound.
Continuity mistake: During the final battle, Borodin announces "20 seconds to impact". Ramius and Ryan discuss a book Ryan has written for the full 20 seconds, then Borodin updates, saying "10 seconds to impact" despite the torpedo and the sub both remaining on their same courses at the same speeds.
Continuity mistake: As the helicopter nears the Dallas sub, Ryan is in the right-side doorway at the rear of the opening. A spotter is on the same side of the craft, at the front of the opening. The spotter looks straight out the right side of the chopper and shouts for the pilots to turn left, as he has spotted the Dallas at three o'clock (which is where he is looking). But turning left moves away from the Dallas, not towards it.
Continuity mistake: When Jack Ryan is being flown, via chopper, to meet up with the USS Dallas, the pilot expresses concern for the fuel situation. First we see this pilot in the right-side seat from a good distance behind his left side as he is talking. When he looks back at Ryan in close-up, the shot is backwards so that he seems to now be in the left seat, looking over his right shoulder (notice his name on the helmet is backwards). Then, after the sub is spotted, he speaks again, and is seen back on the right side, looking over his left shoulder at the left-side co-pilot.
Continuity mistake: When Jack is on the plane and when he is walking through the airport with two FBI-looking men, he is wearing a white turtleneck sweater under his trench coat. As he enters Admiral Greer's office and takes off his coat, he is suddenly dressed in a crisp white dress shirt and tie with a suit jacket.