As a student, can I tutor my classmates?
As Kasa stated, this is not misplaced at all and only takes on that appearance because of the formatting of English questions.
As president of the United States, how did Roosevelt handle the Depression?
This is the same structure. It's a tiny bit awkward since it could theoretically be understood as asking a sitting president what his official position is about the job Roosevelt did. There are better ways to ask the same question: Once elected... After taking office... As president...
As a writer, what are some of your favorite motifs?
As a teacher, who was your most memorable student?
If you had a bad teacher who really didn't like you, you could get marked down for this since the clauses can be misunderstood as modifying the question words what and who. (See Kasa's answer, where s/he misunderstands them as modifying your.) Really, though, they're not misplaced at all and modify the entire sentence adverbially as part of your question to the listener. You could add speaking... in your experience... in your opinion... to the beginning to make this more explicit, but it's not necessary.
None of these is an example of a misplaced modifier in the sense that you mean and none of them are wrong, but all of them except the teaching one sound a little more awkward when you say it this way. It's more natural to put that clause at the end of the sentence: Can I do this as a student? How did he handle it as president? What are your favorite Xs as a writer?