If you allow $F$ to be a continuous functor, which it appears you do since localization functors are continuous, then I believe this is true. Corollary 2.1.1 in the following paper seems to be useful:
E.Dror Farjoun, Higher homotopies of natural constructions, Journal of Pure and Applied Algebra, Volume 108, Issue 1, 8 April 1996, Pages 23-34, ISSN 0022-4049,
It says that if $\phi: X \rightarrow Y$ is a map and $G$ is a continuous functor such that for all spaces $A$, the induced map $map(Y,GA) \rightarrow map(X,GA)$ is a weak equivalence, then $G(\phi): G(X) \rightarrow G(Y)$ is a weak equivalence. Letting $G = L_f \circ F$ should give the result you want. The induced map will be an equivalence since $\phi$ is an $f$-equivalence and $L_f\circ F(A)$ is $f$-local for all spaces $A$.