Backed by the Pentagon: The scenes on the flight deck were shot on the actual Enterprise. Scott Glenn (Mancuso) also spent a month aboard USS Salt Lake City where he was treated as though he was the commanding officer. According to director John McTiernan he came back completely different, very soft-spoken and calm, with a manner he described as being similar to a college president.
The Navy was fully onboard for the movie because they were hoping The Hunt for Red October would do for the submarine service what Top Gun had done for Naval aviation.
In a way it worked, the crew of the realUSS Dallas adopted the films tagline, "The Hunt is On" as the boats unofficial motto.
Fake Russian: It's a Hollywood flick, so it's a given that many of the major Russian characters were played by non-Slavic actors. The most obvious is Sean Connery as Lithuanian-born Captian Ramius, who doesn't even try to alter his Scottish accent. Averted, though, with most of the Soviet extras since most had obviously Slavic features.
James Bond is now a Lithuanian nuclear ballistic missile submarine captain. Doubly hilarious because canonically Bond had been aboard a submarine among his service in World War Two, and even makes a comment to the effect of how submarines are growing more luxurious. He now gets to command the world's largest sub!
The Other Marty: Klaus Maria Brandauer was originally cast as Ramius, but two weeks into filming he had to quit because of a serious injury (he broke both his legs in a car accident). He recommended Sean Connery, with whom he starred in Never Say Never Again, to take over the role.
Prop Recycling: The teddy bear Jack Ryan gets his daughter at the end is the same teddy bear John McClane was bringing for his kids at the beginning of Die Hard, also directed by John McTiernan.
Acclaimed Austrian actor Klaus Maria Brandauer was originally cast as Marko Ramius, and participated in the production process to the point of getting costumes fitted and attending several rehearsals. And then he was in a car accident and broke both of his legs. It was at Brandauer's suggestion that Sean Connery was cast to replace him. Had Brandauer been able to finish the film, it would have had a very, very different feel to it than the actual finished product.
Kamarov boasts about being able to fly a plane with no windows in the Alps if he has a stopwatch and a compass. This is actually a real navigational method known as Dead Reckoning. If you know how fast you are going, what direction you are traveling in, and how far you need to go to get to a particular point, you can indeed use a compass and a stopwatch to reliably get there, barring unobserved changes in weather. You wouldn't normally use it to navigate through a canyon though. Submarines, for obvious reasons, use this technique far more often than airplane pilots, as submarines lack windows to see outside with.