 Projective direct image need not preserve surjections

For $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$, let $D$ consist of $0$ and infinity. Its ideal sheaf is $\mathcal{O}(-2)$, giving an exact coherent sequence $0\to \mathcal{O}(-2)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$. The map on global sections is the diagonal $k\to k\oplus k$, which is not surjective. Since direct image to the point is global sections, projectivity does not make direct image right exact.