In October 1899, Doctor Worley says that it will soon be 1 January 1900, and a new century. In fact, the transition between the 19th and 20th centuries did not happen until 31 December 1900 to 1 January 1901.
Tick-Tock's thinking mechanism is activated by winding the key under his left arm, and his talking is activated by winding the one under his right arm. However, when he asks Dorothy to wind his thinking key before entering the ornament room, she winds the one under his right arm.
When Dorothy first meets Tick Tock, he tells her that he will be her humble servant, takes off his hat and bows before her. When he's bowing if you look closely you can see the "puppeteer" crouched behind him.
When Dorothy is locked in the tower room in Mombi's palace, the view reveals that Oz is flat and the Deadly Desert can be seen plainly. In the Wizard of Oz (both the book and the 1939 film), it's implied that Oz is very hilly and mountainous and the edge of the country is a long way from the Emerald City, too far to be seen at such a height.
The melting point of silver is 961.78°C, or 1763.2°F. Its boiling point is 2162°C, or 3924°F. Therefore, Dorothy would not have been able to drink "hot liquid silver", or even hold the cup for that matter.
When Dorothy steals Mombi's ruby key, a headless Mombi is heard snoring; snoring occurs from the flow of air causing the soft palette at the back of the throat to vibrate, therefore without a head, Mombi would not snore.
Dr Worley and Nurse Wilson apparently did not think it through that if Dorothy did get brain damaged by surgery Aunt Em would find out even if they locked Dorothy in the cellar with the other patients.
Jack Pumpkinhead tells Dorothy that Mombi hasn't remembered that he's in the tower because she hasn't worn the head that she was wearing at the time since then, but the head in the cabinet that Dorothy opens isn't the one that Mombi was wearing when she saw Dorothy. How does it recognise her?
The goof items below may give away important plot points.
After Mombi switches her head to the brunette (head #4) her lips are out of sync while telling Dorothy about the Scarecrow, clearly indicating that it's a voice over by Jean Marsh, the original nurse/Mombi.
After Dorothy uses the ruby slippers to escape from the Nome king's mountain and restore the emerald city, the Gump begins eating the grass, even though he insisted earlier that he doesn't have a stomach for it to go to.