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I would guess that the compact spaces you are looking for are extremely rare.

For example the extremely simple contractible space $I=[0,1]$ is not suitable:
Consider the inclusion $j\colon U=(0,1)\hookrightarrow I $ and take for $F$ the sheaf on $I$ defined by $F=j_!(\mathbb Z_U)$, the constant sheaf $\mathbb Z_U$ on $U$ extended to $I$ by zero.
We then have $H^1(I,F)=\mathbb Z$, as proved in Bredon's Sheaf theory, page 82.

There is a very similar statement in scheme theory saying that $H^1(\mathbb A^1_k,j_!(\mathbb Z_U))=\mathbb Z$, where now $U$ is the complement of two closed points in the affine line $\mathbb A^1_k$: see Hartshorne's Algebraic Geometry, Exercise III 2.1

Of course on afffine schemes, quasi-coherent sheaves have zero cohomology in positive degree, but that is not a purely topological statement and as shown in the example 2) above does not apply to arbitrary sheaves of abelian groups: even theorems by Serre necessitate some hypotheses!