Which sentence has a pronoun-antecedent error?
a) everyone who wants to succeed in life needs to manage his or her money.
b) when e-mail goes down, every one of the teachers holds their breath anxiously.
c) harrison wasn't sure if his work was good enough to pull an a in his biology class.
d) the two men hadn't seen each other since graduation, so they almost didn't recognize each other.

The answer to your question would be that the sentence that has a pronoun-antecedent error is the following one: When e-mail goes down, every one of the teachers holds their breath anxiously. That is, your answer would be B.

The selected sentence is an example of the use of "singular they". Singular they is the use of the pronoun "they" and any of its derivative forms ("them", "their", "theirs", and "themselves") as a singular pronoun. It typically occurs with an unspecified antecedent.