Let $E$ be a sheaf topos $E=\operatorname{Shv}(C)$ and $\{x_i:\operatorname{Sets}\to E\}$ be a set of geometric morphisms $x_i:Sets\to E$ such that the morphism $$ x^*\colon E\to \prod_i \operatorname{Sets} $$ induced by the left adjoints $(x_i)^*$ is faithful (this is saying that $\{x_i\}$ consists of enough points).
The morphism $x^*$ has a right adjoint denoted by $x_*$.
Does $x_*$ preserve directed colimits? If not, what is a counterexample?
No, and there is no reason to expect this anyway. For example, let $\mathbf{sSet}$ be the category of simplicial sets – this is a presheaf topos by definition and so has enough points: just take $x_i : \mathbf{Set} \to \mathbf{sSet}$ to be the unique (essential) geometric morphism whose inverse image functor $x_i^* : \mathbf{sSet} \to \mathbf{Set}$ sends a simplicial set $X$ to the set $X_n$. Thus we have a geometric morphism $x : \mathbf{sSet} \to \mathbf{Set}^{\mathbb{N}}$.
It is well-known that $\mathbf{Set}^\mathbb{N}$ and $\mathbf{sSet}$ are locally finitely presentable (l.f.p.) categories, and it is also not hard to check that a right adjoint between l.f.p. categories preserves filtered colimits if and only if the left adjoint sends compact (a.k.a. finitely presentable) objects to compact objects. Now, $\Delta^1$ is compact as a simplicial set, yet its inverse image in $\mathbf{Set}^\mathbb{N}$ is not compact. (An object $Y$ in $\mathbf{Set}^\mathbb{N}$ is compact if and only if $\coprod_{n \in \mathbb{N}} Y_n$ is a finite set.) Hence $x_* : \mathbf{Set}^\mathbb{N} \to \mathbf{sSet}$ does not preserve filtered colimits, and hence does not preserve directed colimits.
That said, it is not hard to check that $x^*$ sends compact objects in $\mathbf{sSet}$ to $\aleph_1$-compact objects in $\mathbf{Set}^\mathbb{N}$, so it follows that $x_* : \mathbf{Set}^\mathbb{N} \to \mathbf{sSet}$ preserves $\aleph_1$-filtered colimits. More generally, for any Grothendieck toposes $\mathcal{E}$ and $\mathcal{F}$ and any geometric morphism $f : \mathcal{E} \to \mathcal{F}$, there exists a regular cardinal $\kappa$ such that $f_* : \mathcal{E} \to \mathcal{F}$ preserves $\kappa$-filtered colimits. This is because Grothendieck toposes are always locally presentable (but not necessarily l.f.p.) and the accessible adjoint functor theorem ensures that any right adjoint between locally presentable categories is accessible (but not necessarily $\aleph_0$-accessible).