Continuity mistake: On the airplane, Rock Hudson takes a pillow from the stewardess, and puts it behind his head. They cut away to the stewardess, and back to Hudson, and the pillow is gone. They cut to the stewardess again, then back to Hudson, and the pillow reappears.
Continuity mistake: When Wyler and Susan are escaping in the van, the windscreen is shattered and Wyler punches a hole for visibility. In all further shots of the van, the windscreen is obviously intact with the shattering merely cosmetic.
Continuity mistake: When he's called in to see the Captain, Montag wears his wristwatch turned inward (with the face resting over his pulse point). Seconds later, though he hasn't touched it, the watch has rotated itself 180 degrees and is now on top of his wrist. (00:26:05)