cohomology - Cech to derived spectral sequence and sheafification

Yes, this is true in general.

It suffices to show the stalks vanish. Pick $x in X$ and take an injective resolution $0 to {cal F} to I^0 to cdots$. For any open $U$ containing $x$, we get a chain complex

$$0 to I^0(U) to I^1(U) to cdots$$

whose cohomology groups are $H^p(U,{cal F}|_U)$.

Taking direct limits of these sections gives the chain complex

$$0 to I^0_x to I^1_x to cdots$$

of stalks, which has zero cohomology in positive degrees because the original complex was a resolution. However, direct limits are exact and so we find

$$0 = {rm colim}_{x in U} H^p(U,{cal F}|_U) = {underline H}^p({cal F})_x$$

as desired.

Generally, cohomology tells you the obstructions to patching local solutions into global solutions, and this says that locally those obstructions vanish.