When Paul shoves Dandy to the floor, the overhead shots show footprints in the sawdust from prior takes. When Dandy sits up, however, there are no footprints anywhere around him.
When the freaks are getting ready to drown Dandy and he is in the Houdini tank, he looks up at Jimmy when Jimmy is up on the ladder saying he (Dandy) always wanted to be part of the show. It cuts to Dandy's perspective, which looks like he is looking up at Jimmy through water, but they haven't filled the tank yet.
Paul spits on the right side of Dandy's face but when Dandy gets up, he wipes the spit off the left side of his face in one of the opening scenes of the finale of Freak Show.
In the flashback of Massimo teaching Elsa to walk with her prosthetic legs, she drags her feet, and thus extending them, when Massimo helps her stand up after falling. This would not be possible with wooden prosthetics.
When Michael Beck helps Elsa Mars of the floor her legs are not made out of wood.
When Jimmy closes Amazon Eve's eyes, you can see his claw segment move. However, they are supposed to be wooden hands, and so his claw segments would not be able to move.
After the Freaks quit the show, Dandy goes on a murder rampage with a small pistol that holds realistically only about ten rounds. However, he manages to pop off more than a dozen Freaks, which would be impossible with only 8-10 rounds. And he doesn't reload the pistol in-between.
When Elsa waits in the studio you can clearly see when she is standing up that her legs are not prosthetic, her ankle moves and etc..
Most likely intentional, but at the end of the episode, Elsa sings "Heroes" and "Life on Mars" by David Bowie, which were not released until 1971 and 1977, while the episode is set in 1952 and 1960.