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).
Factual error: When Ryan is brought on board the Dallas, Captain Mancuso climbs down from the conning tower, through the hatch next to the control room of the sub. He then walks/runs down two flights of stairs and enters a short corridor where Ryan is now dropped on the floor by the safety diver and another crewman. I am unsure of the exact location of the lower hatch of the escape trunk on a 688 sub, but I suspect it would be on mid deck. When the camera turns to face Mancuso again you see the curvature of the hull next to him, suggesting this takes place on an upper floor on the sub. So why did he run down two stairs to seconds later end up on the top floor of a sub with only three floors inside the pressure hull?
Suggested correction: All escape trunks on all submarines are on the top of the pressure hull. The lower hatch would be a few feet below the top of the pressure hull. The only deck (floor) to be able to reach the escape trunk is the upper level (top floor), so the shot must show the curvature of the hull. If it didn't, then it would be a factual error.
On 688 class submarines such as the USS Dallas, the forward escape trunk lower hatch is immediately aft of Crew's Mess in Forward Compartment Middle Level.
Factual error: In some stateroom scenes in the October the bulkheads have numerous little lockers installed because storage space on a submarine is at a premium. The latches on the lockers are of a unique design created only by USA Navy ship and submarine construction shipyards.
Suggested correction: Given the vast step upwards in technology and the power of Soviet Intelligence at the time, there's no reason why they shouldn't know about something as least classified a latch on a locker.
Suggested correction: The Soviets stole from the US like it was a bodily function, to the point where, in anticipation of such, our engineers would put microengravings mocking them "when are you going to stop stealing" in Cyrillic, on the surface of computer chips et al. The only real objection to the presence of those lockers is that the Soviets would have bigger fish to fry in terms of copying.
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.
Continuity mistake: When Jack is getting off the plane in the beginning of the movie, his jacket collar is up. When he meets the 2 men who are there to pick him up, the collar is now down. (00:06:10)