Let $X$ be a topological space, $\mathcal F$ a sheaf of abelian groups on $X$. Let $\breve{H}^n(X, \mathcal F)$ denote the $n$-th Čech cohomology of $X$ with coefficients in $\mathcal F$. Thus $\breve{H}^n(X, \mathcal F) = \varinjlim \breve{H}^n(\mathcal U, \mathcal F)$, the limit being taken over open coverings partially ordered under refinement.
The construction of Čech cohomology makes sense for a presheaf as well. If $\mathcal G$ is a presheaf, the sheafification map $\mathcal G \to \mathcal G^+$ induces maps $\breve{H}^n(X, \mathcal G) \to \breve{H}^n(X, \mathcal G^+)$. For $n=0$, this map is an isomorphism, on the one hand because $\breve{H}^0(X, \mathcal G^+) = H^0(X, \mathcal G^+)$, and on the other hand because the limit which defines $\breve{H}^0(X, \mathcal G)$ identifies this group with $H^0(X, \mathcal G^+)$ by the usual explicit description of sheafification.
What can one say about the maps $\breve{H}^n(X, \mathcal G) \to \breve{H}^n(X, \mathcal G^+)$ for $n>0$? Are they also isomorphisms? are they injective? surjective? Is there a spectral sequence hiding somewhere?...