Let $\mathcal{G}$ be a Grothendiek topos and let $(\mathcal{C},J)$ be a site for $\mathcal{G}$.
Is it true that if the sheafification functor $$ \mathcal{G} \leftrightarrows \text{Set}^{\mathcal{C}^{\text{op}}} $$ is open then it is an equivalence of categories?!
I have just a very intuitive idea to prove this conjecture. Suppose that $(\mathcal{C},J)$ is a locale with enough points. Call $D$ the trivial topology over $\mathcal{C}$ such that $\text{Set}^{\mathcal{C}^{\text{op}}} = \text{Sh}(\mathcal{C},D)$. Then we are just asking to the identity map $$\text{Points}(C,D) \to \text{Points}(C,J) $$ to be open and continous. Of course now, $J$ must be the trivial topology.
By Lemma 1.4 of Open maps of toposes by P. T. Johnstone (cf. here or on the nlab), the sheafification functor is open if and only if $\mathcal G$ is an open subtopos of $\widehat{C}$, i.e. if and only if there exists a subterminal object $U \to e_{\widehat{C}}$ such that $\mathcal G$ is equivalent to $\widehat{C}/U \simeq \widehat{C/U}$, where here $e_{\widehat{C}}$ is the terminal presheaf on $C$ (cf. also the pertaining Stack Project page here).
Now, a subterminal object of the presheaf topos $\widehat C$ is precisely a sieve of $C$, that is a full subcategory $S$ of $C$ such that if an object $c$ of $C$ belongs to $S$, than any arrow of $C$ with target $c$ is in $S$, as well. Therefore as long as the category $C$ is non-trivial you get a great number of open subtopoi of $\widehat C$, which correspond to the Lawvere-Tierney topologies on $\widehat{C}$ canonically induced by the subterminal object $U$, and they are equivalent to presheaves topoi of the form $\widehat S$, where $S$ is the sieve of $C$ associated to the subterminal object $U$ of $\widehat C$.