Corrected entry: This could be an error done on purpose to simplify the audience understanding. When the Russian Alpha fires on the Red October towards the movie's end, the torpedo's course is stated as 315. Ramius then tells Ryan to steer the sub to a course of 315 to head into the torpedo. That's wrong. If the torpedo were on a bearing of 315, the sub would need to be on a bearing of 135 to be heading directly into it.
Factual error: In the opening scene of the movie, the Red October is being escorted out of the Russian harbour by a United States Coast Guard Cutter and U.S. Navy sea tugs. (00:02:30)
Suggested correction: You may be aware of the fact that they used American boats (I doubt they had access to Soviet ones), but there are no distinctive American markings or livery visible. Most notably, you don't see USCG racing stripe in the boat to starboard (the only one which is not a tug).
Continuity mistake: When Ryan is helo'ed to the Dallas it does not have a DSRV attached, but they use one later in the movie to board Red October.
Suggested correction: In the film more than 24 hours pass between the time Ryan gets on Dallas and when they show the DSRV attached to her. If you remember earlier in the movie it was stated that they could have the DSRV anywhere in the world in 24 hours. Therefore, one may assume that this time was used to get the DSRV to the Dallas.
Dallas was in the middle of tracking Red October and would have to have broken off the tracking (as it did anyway to pick up Ryan) in order for a sub tender to meet Dallas, place the DSRV on her, and then Dallas would have to hightail it back to find Red October. Plus getting the DSRV out to Dallas seems like a critical part of the story. While some scenes end up on the cutting room floor in every movie one has to wonder why there wasn't even a mention about having to surface for the DSRV.
Dallas didn't need to track the Red October anymore because the extraction point was already arranged, so it was very plausible the Dallas could get access to the DSRV.
Corrected entry: As Ryan embarks to shoot it out with the KGB agent down amid Red October's ballistic missile tubes, Ramius warns Ryan to be careful what he shoots at, because the nukes don't "react well to bullets". Of course it's wise to be careful with guns, but you could unload an assault rifle on the tubes and the nukes inside wouldn't care. The tubes are tough enough to stand up to launching their rockets. A stray bullet would be nothing.
Correction: Ramius actually says: "Most things in here don't react well to bullets". He wasn't specifically referring to the missiles. There are many things on board a nuclear sub that could be damaged by gunfire.
Correction: The rockets are not fired in the tubes. The missile is expelled from the tube by compressed gas. The rocket is only fired when it is safely clear of the submarine.
Correction: The torpedo fired at Red October is reported at bearing 315. Then Red October seers a course of 315. That is accurate to set a collision course with the torpedo.
How? Both objects would then be on a heading of 315 thereby going in the same direction, not heading into each other. If the torpedo was traveling on a bearing of 315, the reciprocal heading would be 180° opposite, or 135.
kaevanoff
When they detect the torpedo the Russian says there is a torpedo in the water bearing 315, meaning the torpedo is 315° from them, not that its course is 315. Ramius then orders the sub to head in the direction of 315 which is the direction they detected the torpedo as coming from.
jimba
The torpedo's bearing is 315 from the sub's position, but has a COURSE of 135, towards the sub.