If there were a proof to show that P=BPP, how would this affect zero knowledge proofs and the fiat-shamir heuristic for Turing an interactive proof into a non-interactive proof?
It would not affect this at all. Note that under reasonable complexity hardness assumptions, it holds that P=BPP. So, even though we don't know how to prove this unconditionally (and in fact seem far away from doing so), it is what we assume to be correct.
I am not sure why you would think that this would affect the Fiat-Shamir heuristic. Even if P=BPP it doesn't mean that randomness has no role. In particular, you cannot sample a distribution deterministically. P=BPP relates to the ability to decide a language, not sample distributions. In particular, the necessity of randomness for zero knowledge holds, even if P=BPP.