Deliberate "mistake": Throughout series: Five-O was said to be Hawaii's state police force, and was shown headquartered in the Iolani Palace. For decades, tourists were disappointed to learn that in reality, Hawaii had no state police, and that the palace was a state museum, not a state police station.
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Trivia
Sunburned actors were a frequent problem for Hawaii Five-0's make-up crew. So many guest stars arrived in Hawaii early and sunbathed before reporting to work, the make-up team had to devise a special formula to neutralize red skin for the camera. Dubbed "mellow yellow," it remained a Five-0 make-up staple for all 12 seasons. See more...
Updates: 
Series 4
Across whole show
Audio problem: Throughout series: "Hawaii Five-O's" favorite sound effect was squealing tires, and the effect was nearly always added whether McGarrett's big black Mercury actually "peeled out" or pulled away at a normal speed. More than once, the tire squeal sounded when the car was sitting on beach sand. Nice trick - if you could do it. But alas, you can't "burn rubber" on a beach.
Continuity: Throughout first season: Due to stock footage spliced in from the pilot film, McGarrett's '68 4-door Mercury sedan often peeled away from the Ionali palace and then turned into a '67 2-door coupe as it cruised down Honolulu's streets.
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