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).

I have no idea about its credibility, and I'm certainly not trying to discourage Zhang's or Zyskowski's out there. But you might find this article interesting:

L. Einav, L. Yariv. What's in a Surname? The Effects of Surname Initials on Academic Success. Journal of Economic Perspectives, 20 (2006), 175-187.

We present evidence that a variety of proxies for success in the U.S. economics labor market (tenure at highly ranked schools, fellowship in the Econometric Society, and to a lesser extent, Nobel Prize and Clark Medal winnings) are correlated with surname initials, favoring economists with surname initials earlier in the alphabet. These patterns persist even when controlling for country of origin, ethnicity, and religion. We suspect that these effects are related to the existing norm in economics prescribing alphabetical ordering of authors’ credits. Indeed, there is no significant correlation between surname initials and tenure at departments of psychology, where authors are credited roughly according to their intellectual contribution. The economics market participants seem to react to this phenomenon. Analyzing publications in the top economics journals since 1980, we note two consistent patterns: authors participating in projects with more than three authors have significantly earlier surname initials, and authors writing papers in which the order of credits is non-alphabetical have significantly higher surname initials.

It's absurd and ludicrous to take this kind of bias into account when choosing your advisor, though...

Based on my own experience, even if your last name comes before your advisor's, your own research community will regard any joint work with your advisor to be primarily your advisor's work, despite your advisor's protests to the contrary, until you start publishing independently. The Matthew Effect is a much more significant than your position in the alphabet.

(My name comes before my advisor's, and I've advised students with names before mine and others after mine. I work in a field that orders authors alphabetically.)