When the dignitaries visit the complex, Maggie mispronounces the word "seismographs." She says "SEIZE-mographs" instead of the correct pronunciation, "SIZE-mographs." As an expert holding a doctorate in geology, should ought to know the correct pronunciation of such a basic term.
After the dive in the mini-sub, Dr. Sorenson asks Dr. Rampion for the negatives so he can inspect them more carefully. The negatives are 120 square format negs, but the cameras used onboard the sub were 35mm format.
Tanganyika (where main events take place) existed from 1961-1962, then became the Republic of Tanganyika from 1962-1964 before changing its name to the United Republic of Tanzania after merging with Zanzibar. When this film was released in 1965, the country's name had already changed.
In the beginning, while explaining the plan to the commissioners, Dr. Stephen Sorenson said that they are going to use a ten megaton missile to bore through to the mantle. Later, when Dr. Stephen Sorenson and Dr. Rampion are analyzing two spectrographs, they discover a hydrogen signature. Dr. Rampion says, "But you didn't use a hydrogen bomb." The upper limit yield of fission weapons is about 500 KT (about 1/2 of a megaton). Fusion weapons were designed to release more power than a fission bomb, as much as 50 megatons. The bomb Sorenson used must have been a fusion, or hydrogen bomb.
Most of the vehicles in the movie have their steering wheels on the left; however, the action is supposed to take place in Tanganyika, a former British colony which drives on the left and auto steering wheels are on the right of the car.
Near the beginning of the movie, Dr. Sorenson and Rampion are flying a helicopter to the scene of a huge animal stampede. The low-angle in-flight shot of them in the cockpit shows clouds above them, but they are motionless. This indicates that this scene was filmed on the ground in a stationary helicopter.
After Rampion arrives on the ocean floor in a deep sea vehicle to investigate eruptions caused by the missile's explosion, he uses his camera's flash mode while taking photos of the eruptions. But the eruptions are very bright, making flash unnecessary. Also, the flashes would reflect off the vehicle's window, resulting in prints showing nothing.
The goof items below may give away important plot points.
Even if some type of huge crack physically separated a chunk of the Earth from the rest of the planet, with no solid link, the chunk would not fly out into space. Gravity and centrifugal forces would not change, so the chunk would remain exactly where it was, and the planet would remain mostly intact.
When Rampion and Maggie are evacuating the complex via elevator in the movie's climax, a boulder crashes through the elevator car and stops it dead. But the lighting effect outside the elevator window still suggests the car is moving upward.