Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$. 

>Can we conclude that $R^i\pi_*F(-E)=0$ for $i>0$?

The idea being to somehow use that$-E$ is nef on $E$.