What is the effect of a PhD student dyeing their hair blue?

The general advice is that when you're an undergraduate student, a graduate student not yet on the job market, or when you're a tenured faculty, you can do whatever the hell you want.

The problem is that you are vulnerable when you're in the position to be hired, promoted, tenured, or retained. In those cases, having just one conservative person on the hiring/promotion/retention committee (or at the divisional, full faculty, dean or provost levels) can derail you. In those circumstances, you want to stand out in terms of your research, service, and teaching but to try to avoid or mitigate any areas of friction where and when possible.

Since hair color is easily changeable, if I were your advisor, I would recommend that you dress (and hair color) more conservatively when you go on the job market -- and when you come up for promotion/retention/tenure. I would also recommend you wear shoes at your job interview.

The benefit to risk analysis just isn't in favor of frivolity in these high stakes situations. Your departmental faculty may be 100% behind you and your sartorial style but I've seen faculty lose tenure bids at the divisional, full faculty, and provost level despite department support. I've seen grad students not get hired because they wore a t-shirt to a job interview thinking the institution was a cool, hip place. It was, just not that hip.

At all other times during your career, I think you are relatively free to do what you want within the broader norms of your particular cohort and department.

_{Note that while my home department is anthropology and I'm currently at a R1, I've also taught at two SLACs and have seen enough shenanigans in other departments and at divisional/university levels that my advice is not restricted to just anthropology at R1s but is intended as general advice. Ymmv.}

This definitely depends on environment: your research field's culture, your department's culture and your university's culture.

As someone that had a mohawk phase often during grad school, I can only speak from my time in my PhD program in mathematics (in the USA). In my experience, I would cut it off before any conference, any research visit, and the job market as it felt not right for me. I kept the mohawk when teaching. My university never complained about the hairstyle and I won teaching awards from the students. The most I heard from colleagues was that it probably made mathematics "more relatable" to the students and the occasional "you should grow a tail in the back so you're like that jedi...." On the other hand, I would feel uncomfortable with a mohawk in my new university.

In my experience, computer science and mathematics seem culturally the least focussed on appearance. Everything is contextual, though.

In some fields with significant client-facing time, colored hair (or other similar notable features like significant visible tattoos or piercings) is generally unacceptable. My knowledge of this primarily comes from clinical psychology, but I'm sure that there are other similar fields (for instance, social work). For somewhat obvious reasons, maintaining a professional and somewhat conservative appearance is important when a significant part of your degree involves doing clinical work.

In non-clinical fields, I think that this is generally okay (unless you have some particularly conservative faculty), but I will let other answers address that.