"Do you know who the new principal is?"
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Because it is asking a yes or no question!
Do you know who the new principal is? 你知道新校長是誰嗎 Yes/no 知道/不知道
He is not asking who the new principal is (this would require some kind of description or name), so if he were asking this, it would be 呢), rather he is asking if the person knows who the new principal is. He either knows or he does not know, therefore it is 嗎.
"你們的新校長是誰(呢)?" would be fine, but the nuance is different.
I think 谁 refers to the teacher (who is the teacher) and 吗 refers to 你 (do"you" know the name). There are two questions in the sentence. I would accept both 谁 and 吗。