Čech cohomology of an affine cover can differ from sheaf cohomology (${\check{H}}^1(\{X\},\mathcal{F})=0,H^1(X,\mathcal{F})=k(t)$)

On $X={\mathbb{A}}_k^1$, let $K=k(t)$ and let $\mathcal{K}$ be its constant sheaf. For the two closed rational points $0,1$, take skyscraper sheaves $S_0,S_1$ with value $K$, and put $\mathcal{F}=\ker (\mathcal{K}\to S_0\oplus S_1)$. The maps on the two stalks are identities, so this is a sheaf surjection. All three displayed outer sheaves are flasque. The induced global map is the diagonal $K\to K^2$, so $H^1(X,\mathcal{F})=K^2/\Delta K\simeq K$. The one-member affine cover has zero positive Čech cochains. This even gives an ${\mathcal{O}}_X$-module example: let regular functions act on all copies of $K$ by multiplication as rational functions, not by evaluation at the closed point. The kernel is not quasi-coherent.