Let $\mathcal G$ be a sheaf on a topological space and $X$ (say a sheaf of sets) and suppose its support is contained in a subset $Z$ of $X$, i.e. $\operatorname{Supp} \mathcal G \subset Z \hookrightarrow X$. Let $i$ denote this inclusion, $i_*$ the direct image sheaf and $i^{-1}$ the inverse image sheaf.
I was able to show that $i_*i^{-1}\mathcal G \cong \mathcal G$ for $Z$ is closed by proving that the stalks are isomorphic. A key step in my proof, which would not have worked so easily if $Z$ were arbitrary, was in proving that $(i_*i^{-1}\mathcal G)_p$ is the terminal object $T$ for $p\notin Z$ (obviously, $\mathcal G_p = T$ for $p\notin Z$). I needed to use that $Z$ was closed to pick a neighborhood $V$ of $p$ disjoint from $Z$ (and hence disjoint from $\operatorname{Supp} \mathcal G$) so that $i_*i^{-1}\mathcal G(V) = i^{-1}\mathcal G(\{\}) = T$.
My question is: can we get a counterexample to the isomorphism $i_*i^{-1}\mathcal G \cong \mathcal G$ when $Z$ is open and not closed (or when $Z$ is arbitrary)?
I think that a counterexample, if it exists, would have to involve a non-Hausdorff space, because if $Z$ is open, the only stalks that could be a "problem" would be the ones at the boundary $p\in \partial Z\cap \partial \operatorname{Supp} \mathcal G$. For such points, $(i_*i^{-1} \mathcal G)_p$ is the colimit over a directed system $\{\tilde V_i\} := \{Z \cap V_i\}$ where $V_i$ are neighborhoods around $p$. In a Hausdorff space, $\cap \tilde V_i = (\cap V_i) \cap Z = \{p\} \cap Z = \{\}$ the empty set. So I'm tempted to say that this colimit is the terminal object, for the any sheaf takes the terminal object on the empty set (but that's not true since a sheaf functor does not have to commute with colimits). On the other hand, that's what we expect since $\mathcal G_p = T$, for $p \notin \operatorname{Supp} \mathcal G$.
Let $Z$ be an open set and define $\mathcal G$ to be the direct image under the inclusion map of the constant sheaf $\mathbb Z$ on $Z$. Then $Z=\operatorname{Supp} \mathcal G$ and for some $p \in \partial Z$ we have that the stalk $(i_*i^{-1} \mathcal G)_p$ is the colimit over $\mathbb Z$. This is $\mathbb Z$.