Early in the movie, Oliver Lindenbrook speaks of the "stars and galaxies of outer space." In the 1880s, however, our Milky Way galaxy was believed to constitute the entire universe. Knowledge that other galaxies exist beyond our own did not come about till the 1920s. Thus a man of the 1880s would not use the word "galaxy" in its plural form.
When Alec walks alone over a rock bridge, a dislodged chunk falls away silently. When Alec tries to backtrack over it, two more chunks break away, but this time both pieces make splashes as they hit a phosphorescent pool distantly below.
A Swedish scientist is named Goetabaug. Göteborg is the name of Sweden's second biggest city, but is unknown as the name of a person. The writers apparently assumed that since English-speaking people such as Jack London and Michael York can be named for large English cities, then Swedes can be named after Swedish cities, but this is simply not so.
When Hans is helping Carla and Lindenbrook out of the cave filling with water, from above Lindenbrook is seen with his head to the top of the hole, the water up to his neck. In the next shot from below, the water is only up to Lindenbrook's waist.
The party is underground for many months, yet the men's hair length never changes, and Lindenbrook even comments when the updraft near Atlantis is bothering Carla by making her long hair blow into her face, "If we had shears, we could cut if off." There was no way the men could give each other haircuts, yet their hair never seems to grow throughout the many months they are underground.
When Carla discovers the original three notches and the trick that that Count pulled on them, all of the original party except Alec are at the juncture of the two trails. Sir Oliver chisels a piece of stone from an opposite wall for a specimen, and a huge flood is released. But instead of going down the right trail (or even the left one where Alec was) they take another trail that wasn't there before and wind up in the dead-end room that fills with water.
Hans pursues Count Saknussemm after killing Gertrude the duck. When Hans looks up to find Saknussemm carving into a wall. That shot reveals a camera and/or lighting equipment in the upper-left corner of the shot.
When on the raft, Professor Lindenbrook remarks that their gold is being drawn away by a great force, and says there must be "a conjunction of magnetic forces from the north pole and the south pole". Even assuming that he is talking about the magnetic rather than the geographic poles, such a location would result in the magnetic pull from the poles being cancelled out. In any case, there would be no effect on the gold, as it is not attracted by magnetism.
As the raft is caught in the whirlpool, a thunderstorm, complete with rain and lightning, breaks out. Even given the fantasy aspects of the film, there is no possibility that a thunderstorm, which requires an active atmosphere with clouds, could occur in the center of the Earth.
When Lindenbrook and Count Saknussemm come back out of the sea after escaping the giant lizard, their footprints in the sand from running out during a previous take are clearly visible. It is not their footprints from running in.
When the altar stone is being pushed upward in the volcano, the distance shot of the model shows it remaining in a fixed position. But in the close-ups of the actors, the stone is turning and moving back and forth as it travels upward.