Deliberate mistake: When Alexander is travelling to the future for the first time, it shows a speedy acceleration of the events around him. However, the planes that are seen flying by are moving at normal speeds, or at least speeds way too slow since they should be moving too fast to even see, since skyscrapers are being built in a matter of seconds.
Factual error: When Alexander makes his first trip into the past, you can see the second hand of the pocket watch turning backward. The hand is jumping second by second. This is a mistake. A quartz watch (battery powered watch of our times) would do it this way, but a mechanical one from 1903 would turn backwards the same way it turns forward: 5 steps per second, which for the untrained eye looks like a smooth movement.
Factual error: In the middle of the movie, when Alexander is traveling through time and the landscape around him is evolving, the geologic sequence makes no sense. It shows sand and dust covering what was once New York, then water forming into a river and wearing down the gorge, and then a glacier covering the area. The glacier would have exerted hundreds of millions of tons of pressure on the land, smoothing out the gorge that had worn down through the sedimentary rock. The glacier would then have melted into the hollow it had formed, much like the Great Lakes. It would have made more sense for the glacier to have come first, then retreated and melted to form a river that carved out the gorge.
Continuity mistake: When Alex uses his first time machine to travel into the past, one can only assume he follows the same procedure that is shown when he travels into the future for the first time; it is simply not shown the first time to inspire awe and suspense. That said, when Alex is finished traveling to the future for the first time, both smaller spinning discs do not fold outwards as they were before it's use. Before he uses it to travel to the future though, you see the shot of the machine's light being wrapped around the machine as the discs gain speed. If the discs did not fold out when done traveling into the past, then they would not have done so after he traveled into the past. He obviously did not set them back either, because he can use the machine again without doing so.
Plot hole: When Alexander is traveling into the future and goes through the ice age he is visibly affected by the extreme cold temperature outside the time bubble. However when he stops in the future when the moon had been destroyed then continues he is not affected by the explosions and extreme heat around him. Based on the effect the extreme cold had on him (frost on his face and hair) he should have been roasted alive.
Revealing mistake: When Alex uses his time machine for the last time to destroy the Morlocks, watch the closeup of his machine when it's activated. In the lower left hand side of the machine you'll see Alex's laboratory in the background. This is confirmed by director Simon Wells himself in the DVD commentary.
Continuity mistake: When Alexander drops and grabs the necklace, his hand sticks out of the time machine. His nails grow. In many scenes after that his nails return to normal.
Revealing mistake: During Alexander's long journey, it shows an elaborate geological sequence. It shows the area turning to desert and then trees and plants begin to grow. If you look at the background, trees grow up, then shrink in the same manner instead of dying and falling over. The special effects crew just reversed the growing sequence.
Continuity mistake: The operation of the Time Machine has no continuity. In the beginning of the film, both when he goes backward and forward in time, Alexander has to activate the machine and it sort of warms up: The glass blades spin and open outward, the cones of light open up in sequence, and the light bubble slowly envelopes the machine. But when he is escaping the destruction of New York and when he fights with the Morlock, both times he just throws the lever forward and the machine speeds off with no preliminary activity.
Visible crew/equipment: In the widescreen version, when Alexander stops his time machine in the year 2030, he lowers the steps to get off. As he's doing this you can see the reflection of a few crew members in the pressure gauge located by the steps.
Revealing mistake: When Alexander is about to wake up at the end of his journey from 2037 to 802,701, it cuts to a shot of the year display counter on his machine. The 100,000's digit is spinning just as fast as the 1's digit and all the others, rolling over several times without the millions digit moving. If the counter was moving correctly, as it is in all other shots of it, then the 100,000's digit should be moving slowly, or the millions digit should also be moving.
Continuity mistake: After the final fight is over, most of the blood on Alexander's face, and especially the blood in his mouth, disappears.