At minute 16:50, the sub being shown underwater as the Tiger Shark is of a completely different profile and conning tower arrangement. The submerged sub having four antennas and the surfaced sub having only two.
The Tigershark appears to have pierced the saucer at a 30° angle, and the subs bow inside the saucer shows a 45° angle, yet all of the interior shots back inside the submarine are perfectly level.
At the end of the movie Holloway and Neilson walk outside during the day talking about the future. Neilson looks up at the sky and it is not a bright sunshine sky, but a night sky full of clearly visible stars.
The Magnetic North Pole is not located under the Geographic
North Pole, which is where the UFO returns in the film to "recharge its batteries" from the Earth's magnetic field. Science had acknowledged this at least two decades before 1959, which was when the film had been released. More current maps show it shifting a bit to the north of Canada.
While venturing out to inspect the exterior of the submarine for damage the frogmen would not survive the arctic water in clinging wetsuits. They would need bulky dry suits to insulate themselves from the near freezing water.
There is no Nobel Prize in oceanography.
At minute 30:50, stacks of dishes and cups are shown on an open shelf in the Officers Mess. Submarines, and also most surface vessels do not use open shelving due to rough seas and even typical steering maneuvers which would leave a pile of broken cups and plates on the deck.
Just as the saucer breaks through the ice, you can briefly see the rod with which the model is being pushed up.
At minute 44:20 when the Skipper tells Reef to make all engines full stop, the model being used in the underwater scenes has the propellers running in forward, rather than reverse as previously ordered, to extract the Tiger Shark from the saucer.
At the very beginning, the narrator is talking about the North Pole but there are snow covered mountains in the shot as the jets fly over. There are no mountains at the North Pole.
After the Tiger Shark rigs for "silent running," they continue to communicate over the loud speakers.
The alien saucer has enough advanced technology to make ships and submarines disappear in the Arctic Ocean, can deflect or freeze torpedoes fired at it, speak telepathically to humans, yet the cyclops has no defense against being shot in the eye with a pistol.