Errors in geography: The Sultan of Basra's palace has many decorations in the form of Hindu iconography, which would be found in India, but not in the ancient Middle East. (It was considered "Oriental" enough to pass by a 1940s audience, but stands out for a modern audience.)
Errors in geography: Despite taking place in the ancient Middle East, several scenes show local people speaking in Hindi, which is spoken in India. (Again, it's "Oriental" enough for a 1940s audience not to notice.)
Revealing mistakes: While the genie is flying him to the temple, the platform that Abu is standing on is visible in some shots.
Errors made by characters (possibly deliberate errors by the filmmakers): Almost all of the characters mispronounce "Allah". The first syllable rhymes with "dull" or "null", but in the film, most pronounce it as rhyming with "pal" or "gal."
Continuity: When Ahmad is to be executed, his head is held down with a rope to steady it. But when the executioner raises his sword the next shots of Ahmad show that the rope has disappeared.
Anachronisms: Abu throws tomatoes at his pursuers as he flees though Baghdad. These would have been unknown on Arabia at this time as they originated in the Americas.
Continuity: When the third wish is made and Ahmad disappears, the All-Seeing Eye which was in Abu's hands disappears as he searches for Ahmad.
Incorrectly regarded as goofs: Abu, who we know to be a Muslim, is shown requesting and eating sausages, even though the consumption of pork is very strictly forbidden in Islam. However, the sausage may not be made from pork.
Anachronisms: As the Sultan of Basra flies over the city on his mechanical horse, power or telegraph lines are visible in the background.