Plot hole: The elevator to enter the lab requires a visible-light thumb scan, but Caine enters the lab (and apparently not by following someone) after he has become invisible. A fingerprint is not sufficient for the type of scanner shown. The gloves he wears would also not follow the contours of his fingertips closely enough to copy his fingerprints.
Deliberate mistake: Elizabeth Shue uses a drawer handle as an electromagnet. OK if it's really a magnetic material but lab furniture is generally made of aluminium or stainless steel which is not. There is no way that they would use an iron handle in a potentially wet, and hence corrosive, environment such as a freezer.
Other mistake: Elisabeth Shue uses the blowtorch to set off the fire sprinkler in the hallway, but later there is a raging inferno in the lab with all the important data and equipment and there's not one sprinkler, nor any alternative "electronic equipment friendly" automated fire suppression mechanism such as Halon gas.
Plot hole: Caine's skin becomes visible when it chars, for no reason consistent with the other invisibility. Further, it drops off in an impossible way (since there's no indication the invisibility otherwise changes metabolism/healing). Caine acts far too healthy for someone so severely burned. Finally, the extreme damage to skin and muscles that this implies is not seen when he becomes visible. (Even if the material flaking off is clothing, most of the arguments still apply; burned clothing would adhere to burned skin, and would provide little to no protection from the fire.).
Deliberate mistake: When Sebastian goes to his employer's house to kill him, the man is standing by his pool smoking a cigarette. When the boss blows smoke in Sebastian's direction, Sebastian appears and scares him. Since Sebastian is invisible you shouldn't be able to see his pupils, only an outline of his eyeballs, but you can see the pupils.
Factual error: The speed used to administer the serum to Sebastian is not close to the reality. At that speed the vein would have exploded.
Continuity mistake: In the end of the movie you can see that the doors of the elevator are almost closed. But in the next scene when Linda shoots him with the flamethrower, the doors are open again.
Factual error: Duct tape would not stick to bloody skin at 40 below zero, and probably not at higher temperatures either; again because of the blood.
Factual error: The braking system on lifts (elevators) is designed to stop a free-falling car safely and quickly (using gravity and a wedge system). It would not take a long time to stop the car or break up as shown in the film.
Continuity mistake: When Dr. McKay pushes open the freezer door, she is standing behind the row of wires still connected to the door. A moment later, without moving, she is on the other side of them.
Continuity mistake: When the redhead spilled the blood all over the floor to see if he was coming at her there is a little area to the left of the screen he could walk on without her seeing it. The next shot it's gone.