Since there is an answer to the question, I think I should write it down.

There is a simple counterexample to my question: Let $M=N=sSet$ the standard model category of simplicial sets. Let $ex^{\infty}:sSet\rightarrow sSet$ the fibrant replacement functor. It is simplicial as it was noticed in the comments. $Ho(F)$ is well defined and induce an (auto)equivalence of the homotopy category $Ho(sSet)$. On the other hand $\pi_{0}(F)$ is clearly not an autoequivalence of $\pi_{0}sSet$.