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Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?

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    $\begingroup$ Take an elliptic curve $E$ and a degree $0$ coherent sheaf of the form $\mathcal{O}_E(p-q)$, where $p$ and $q$ are two distinct point on $E$. $\endgroup$
    – Francesco Polizzi
    Commented Apr 27, 2015 at 18:13

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What about the "Moebius" local system over $S^1$ with fibers $\mathbb{Q}$ and monodromy $-1$? (It has $H^0=H^1$ by Poincare duality with coefficients).

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