When Radtchenko kisses his girlfriend goodbye he tries to catch what appears to be a Soviet GAZ-66 4x4 utility truck. The production of GAZ-66 did not start until 1966, 5 years after the events of the movie take place.
The Champagne bottle that fails to break when K-19 is christened is labeled (in Russian) as Soviet Sparkling. The trademark "Soviet Sparkling" has not been registered until 1969 when it started selling in the West and was not used internally until 1990s. A bottle made for Russian market would have been labeled as "Soviet Champagne".
At the beginning of the movie, Vostrikov joins naval base by military 3/4 Ton truck. This vehicle is a Dodge WC56 Command Car used by US Army during WW2. Although Soviet troops used a lot of lend-lease US trucks during the War (when both countries were allied), they no longer used it in early 60's.
The sound the camera makes when the group shot of the crew is taken on the ice is that of a fixed shutter, but the camera being used (either a Leica or a Russian copy) would have had a focal-plane shutter.
When the Captain goes into the radioactive reactor room to drag out the last man who was working on the welding, he doesn't close the hatch door to the reactor. This would severely increase the amount of leakage.
When the submarine is leaving the berth, and the two captains are on the tower, the nearby flag blows away from them in the close-up shots. In the long shots, the flag (and other smoke) blows the other direction.
When the K-19 is preparing to leave her berth for the first time, the order "single up all lines" is given. This order means that one set of lines is removed, but the boat should remain secured to her berth with a 'single' set of lines. When the order is given, the crew removes both sets of lines, and the K-19 departs.
When Alexei Vostrikov arrives to the meeting with the Admirals his rank up to this point is that of Captain Second Rank. However, during the meeting he wears the shoulder insignia of a Captain First Rank. In the next segment when he boards the K-19, he once again is a Captain Second Rank.
At the beginning of the movie, when the radio officer checks for Moscow's confirmation during the drill, the close-up shot of the radio panel shows the green light active and two white lights active. A second later the radio panel is shown again (which should look exactly the same as no actions were taken), this time the two white lights are inactive, and a third light is active.
At the beginning where Capt Vostrikov is buttoning his dress jacket, he is wearing full medals indicating a full dress uniform; his rank is that of a Capt 2nd Rank. When he arrives at headquarters in the briefing, his rank has changed to that of Capt 1st Rank but his uniform is now the standard dress using ribbons vs. medals. Full dress would not be used for travel at any rate.
When the submarine is taken down to 300 Meters during a test dive, prior to the missile launch, we see the side of the sub beginning to crush in from the pressure where the large numbers are located on the hull. Yet these large dents are not seen anywhere else throughout the rest of the movie.
At the beginning of the film, when Polenin is discussing the status of the submarine's construction with Vostrikov, he refers to the boat as "she". Russians use the male pronoun when referring to sea vessels.
Reactor Officer Vadim Radtchenko says that the accident may trigger a "Thermonuclear" explosion. This is impossible because Thermonuclear explosion is a nuclear Fusion (fusing Hydrogen atoms into Helium and thus releasing energy) and obviously the nuclear reactor in the submarine is based on nuclear Fission (breaking Uranium atoms thus releasing energy).
Close-ups on some control consoles show new paint over layers that have been severely chipped, not something one would find on a new boat. However this is based in Soviet Russia it cannot be assumed the submarine/boat is new at all. Also Soviet Russia was known for its lack of equipment and 'damaged' goods.
The US Navy helicopter flying around the K-19 is a Sikorsky S-58T, which had its first flight on 19 August 1970, about 9 years after the incident had taken place. In fact, this is not a Sikorsky S-58T but is a UH-24 which is a variant of the S-58T. The S-58T is the Civilian designator for this bird. Also, the S-58T ended production in 1970. It first entered service in 1954 so therefore COULD have (and actually Would have) been the helicopter used to scout the K-19.
When the submarine has broken through the ice, as well as in the other outdoor scenes, the sun is high in the sky. This close to the North Pole, that indicates that it occurs in summer. However, in the simultaneous scenes from the launching and from the High Command inside Russia, it's snowing, indicating winter. If it was winter, there would be constant darkness where the submarine operated.