Not if Pike believes in her and she has his blessing. Then she's been propped up and Pike's intact.
You're looking at it the wrong way. It's not Pike vs. Burnham. That's not the rivalry. It's Burnham vs. Spock as "siblings" and Burnham vs. Saru for the chain of command.
This is what I think will happen (I don't like it, but just making an educated guess). Pike will have some association with Section 31, and that's how they will fold the Georgiou/31 plot in with the main plot.
I'm expecting a smug scene like this between Burnham and Pike:
Burnham has already been written as the righteous moral authority against Lorca, Cornwell and Mirror Georgiou. I hope they subvert expectations with Pike, but I doubt it. Unless the new showrunners secretly want Pike as the heroic lead of this series instead of Michael Boredom, unlikely.