Continuity mistake: When Shaggy and Scooby are performing the barber scene at the Coliseum, Shaggy is holding a sundial in his left hand and you can see that behind him there is a barber's chair, but when we see Shaggy holding the sundial in the next shot, the background has changed to a brick wall.
Continuity mistake: When Shaggy, Scooby and Scrappy have finished perming the lion's hair at the Coliseum, Scooby grabs Scrappy and sends the lion for a spin in his chair. In the next shot, the emperor says "Get off with the lion!" and he pulls a leaver releasing the trap door. But when we see saw the lion falling through the trap door, the barber's chair is nowhere to be seen.
Plot hole: During the diving contest Tiger Morris, who's posing as the sea monster, fakes his cries of terror and flees off to the left of the ocean. Then, seconds later he's suddenly back as the sea monster. Impossible for him to get into disguise that quickly.