The coins the Illusionist gives the poor children each have a ring surrounding a differently colored disc. The technology to make those coins is only a few decades old. They're Czech 50 heller coins, introduced in 1993.
As Eisenheim puts stacks of notes into his leather briefcase at the bank, a silver zipper is visible on the bag. Some early prototype of zippers did exist, but were not in common use well into the 20th century.
When Inspector Uhl finds Leopold to tell him of Sophie and Eisenheim's relationship, Leopold asks "What where they doing? Touching? Kissing? Fornicating?" His face is only shown when he says "fornicating", but he mouths a completely different word. (see Trivia)
Near the end of the film, Chief Inspector Uhl is in Prince Leopold's office. In the close-up scene when the two argue, you can clearly view the boom mic, going from one character to the other, in the reflection on the brass table lamp.
When Uhl opens The Orange Tree folder in the theatre and sees the plans for the locket for the first time, there is a hole in the center of the sheet of paper. When Uhl is in the stables and looks at the plans a second time, the hole has disappeared.
(at around 20 mins) When Inspector Uhl is inspecting the props backstage, Eisenheim has his arms at his side during their conversation. Then suddenly he is facing the inspector at a different angle and his arms are crossed.
Towards the end of the film, the inspector is seen with the necklace having a gold chain. During the flashbacks, Sophie is also seen planting the same necklace with the same gold chain; however, when Eisenheim is giving the necklace back to Sophie at the very end, it is seen having a brown string.
At around the 20:17 mark, When Inspector Uhl is inspecting the props backstage, Eisenheim has his arms at his side during their conversation. Then suddenly he is facing the inspector at a different angle and his arms are crossed.
Duchess Sophie von Teschen reveals to Eduard that Crown Prince Leopold intends to win over the Hungarians by aligning himself with her family. In reality, the Duchy of Teschen was part of the Austrian half of the Empire, so even if the Crown Prince had married her, it would not have had that much of a impact on the Empire's Hungarian citizens.
When Inspector Uhl is investigating Eisenheim's latest illusion of summoning spirits, one of his aides shows an early turn-of-the-century movie projector portraying a color/sepia-based image of a person. Turn-of-the-century film was often hand-colorized using stencil methods. Georges Méliès's fantasy shorts used that method. Several genuine color film technologies were also in use by the early 1920's.
The blood from Sophie was on the right side of the horse yet her wound is on the left side. But, as she rides the horse out of the stables, she is slumped over the right side, her left side pressing against the right side of the horse.
When giving a private exhibition to the Crown Prince and assembled nobles, Eisenheim explains the legend of King Arthur and the Sword in the Stone, referring to the sword as Excalibur. Though popular culture does commonly refer to the sword as such, Excalibur was actually a different sword, given to Arthur by the Lady of the Lake after the sword he pulled from the stone was broken. This is one version - Arthurian legend is hardly 100% factual history, and many incarnations of Arthur indeed conflate the two swords - Excalbur and Claret.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
After the Crown Prince Leopold commits suicide, you can clearly see him breathing as his stomach moves up and down. Even when someone commits suicide by shooting himself in the head, the heart will usually beat for a few more seconds since it is controlled largely by a nervous system internal to the heart. The person is also usually able to take a few more breaths before the entire body shuts down completely as long as the breathing center of the brain is undamaged.