= Solution
For $X=U\cup V$, the <Mayer-Vietoris sequence for sheaf cohomology> is the long exact sequence
$$
0\to H^0(X,\mathcal F)\to H^0(U,\mathcal F)\oplus H^0(V,\mathcal F)\to H^0(U\cap V,\mathcal F)\to H^1(X,\mathcal F)\to\cdots.
$$
We prove the required vanishing by induction on the number $m$ of open sets. The case $m=1$ is an assumption. Put $U=U_1\cup\cdots\cup U_{m-1}$ and $V=U_m$. The induction hypothesis gives $H^p(U,\mathcal F)=0$ for every $p$. The intersections $U_i\cap U_m$ cover $U\cap V$, and every nonempty finite intersection among them is one of the intersections in the hypothesis, so the same induction gives $H^p(U\cap V,\mathcal F)=0$. We also have $H^p(V,\mathcal F)=0$. Exactness of the Mayer-Vietoris sequence now yields
$$
\boxed{H^p(X,\mathcal F)=0\quad\text{for every }p\ge0.}
$$
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