Will it hurt me if my advisor's last name is before mine alphabetically?
Yes, there is a subtle unconscious bias. Even in a field where author names are always alphabetical, papers will be cited in talks as Author1 et al, so if you happen to be Author1 your name will be slightly more disseminated.

Is it true? Yes. Is it fair? No, not entirely. Is it a big deal? No. Should you change your advisor as a workaround? No.

The best way to overcome this bias is going to conferences and getting your face and name known to other people in the field. So it won't matter if your name is Aardvark or Zwingli, because people will know you anyway and know that you did some respectable work.

Another thing you can do to reduce the impact of this bias on your CV is adding a statement on the lines of the following sentence that I put in mine:


  "As is common practice in mathematics, the author order is usually alphabetical and does not reflect a difference in contribution. "


(by the way, I can relate: I have been alphabetically last author on 86% of my joint papers).