Revealing mistake: After the Grinch burns down the Christmas tree in the middle of town and right before they bring out the new one, the Grinch is about to make his escape in the Dump it to Crumpit. The Grinch makes a wide eyed face and you can see he isn't wearing his green and yellow contacts. (00:59:15)
Revealing mistake: When the mayor has his head shaved by the Grinch, Clint Howard grabs the clippers and shaves his own head down the middle. His hair is yanked off his head before he even moves his hand. Even better in slow motion.
Revealing mistake: When the Grinch is powering up his sled, he's turning on switches. The circular display is dark, then turns on during the process. If you look closely, you see the grid and the big black arrow on it before it is lit up.
Revealing mistake: When the Grinch is accepting the Holiday Cheermeister award, Jim Carrey holds up his hand and his palm is facing the camera and you can see the piece of his suit has been slipped over his thumb to make it green and furry.
Revealing mistake: During several of the scenes on the mountain you can tell that the rocks are made of some sort of soft material because you can see them crumple when the Grinch gets near them.
Revealing mistake: As the saw cuts around the Christmas tree, before it falls through the floor, you can see a faint line where the saw tracks around to cut the floor.
Answer: By machine.
The clock in "How the Grinch Stole Christmas" is designed for cinematic effect; a real-world counterpart would rely on precise gearing and engineering to allow three separate wheels to operate in close proximity without interfering with each other. The key would be in the gear ratios and the alignment of the gears to ensure smooth operation of each panel.