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.
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.
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.
Continuity mistake: After the final fight is over, most of the blood on Alexander's face, and especially the blood in his mouth, disappears.