During the escape from the pirate ship, between the first and second musket shots there is footage of the castaways paddling their boat, and in the background a modern ship can clearly and easily be seen on the right side of the screen.
When the group is escaping the island on Nemo's boat, Cyrus cuts the rope tying it to the dock. When he's done, he places the knife in front of him on the raised part of the boat. A few seconds later, the camera pans away and when it returns the knife is gone.
When Pencroff overhears Cyrus and Neb talking about the escape, he has a couple days' growth of beard, but it is close to his face. When Cyrus and Neb chase and catch him outside the tent, the beard is longer.
A high shot of Jane and Helen on the beach shows that the volcano is behind them on the other side of the island. A camera change to their POV on the beach shows the volcano is to the left of the island.
As with most movies depicting giant versions of normally small animals, this one gets it wrong. If the size of an animal is doubled, the surface area of its skin is squared and its weight is cubed. Since the weight of the creature increases by a far larger degree than its other proportions, a giant insect, for example, would be unable to stand or move (this is why elephants' legs are so thick).
Nemo attributes the large size of the creatures on the island to the presence of "Thorium", which, he claims, is found nowhere else on Earth. Thorium is, in fact, a real element, Atomic Number 90, and, far from being rare, is nearly four times as abundant as uranium (nearly as common as lead).
Thorium has no known biological role, and its radiation cannot even penetrate human skin, though it is highly carcinogenic if ingested.
As with most movies and television shows, no one seems to have any conception of how heavy gold is. The treasure involves two chests, one small and one large. The actor varies the effort he uses to move the small chest. At one point he seems to act as if it were more than he could bear, but he is also seen trying to lift and carry the two stacked chests. Gold is 19.3 times heavier than water. That small chest filled with real gold would be too heavy for one man to carry, and a lone individual could not even drag the large chest filled with gold.
The group escapes from the Confederate prison in a hot air balloon, but it doesn't have a burner (something to create hot air and thereby keep the balloon afloat). They also supposedly float 8,000 miles or more, which would be impossible in that type of balloon even if they had a burner.
When the cannon on the pirate ship fires, there are a number of errors; 1) there is no crew loading, aiming, or firing the gun, 2) there are no securing ropes to prevent the cannon from launching itself across the ship, 3) there is no bulwark to which securing ropes could be attached.
In the beginning, when the balloon basket is crashing into the sea, you can clearly see the balloon support ropes are no longer attached to the support ring. Nothing is attached to it anymore. As the support ring breaks, it's also visible it's made of wood painted black, not wrought iron.