Should I put my résumé on LinkedIn when applying for a PhD?
I don't think it really makes any difference.

Admissions committees are not in the habit of doing social media searches for applicants.  In fact, I don't think I've ever heard an academic talk about using LinkedIn at all.  (Maybe it is more commonly used in certain fields that are more closely related to business.)  Even if they did, there's no particular reason they should come across your profile, and no particular reason why finding a resume there or not would influence anyone's decision.

The only possible benefit of posting I can imagine is that if, somehow, one of the admissions committee happened to see it and form some favorable impression of you, that might help their opinion when they read your application.  But that seems like an unlikely coincidence to hope for.

On the other hand, I can't imagine any possible way in which having your resume on LinkedIn could be a disadvantage in the application process.  So from that point of view, you probably might as well post it.

On a related note, I would not recommend trying to add LinkedIn connections to faculty members at the university where you are applying.  Since most probably don't use LinkedIn, this will just result in them getting annoying emails.  And you don't want them to read your application and think "Hey, I remember that name - this is the guy who's sending me all that LinkedIn spam.  What a jerk."