In a sentence, a comma is inserted to represent a separation of the elements in a list or as a marker for the representation of a new thought. In the sentence "Joe isn't allowed to leave work on Saturday [,] until his replacement shows up.", there is one complete thought running through the entire sentence, and therefore no grammatically correct reason to insert a comma.
Your correct answer choice is B: "Joe isn't allowed to leave work on Saturday until his replacement shows up."
Hope that helped! =)