There are several things that one might mean by a “(possibly weak) homotopy equivalence of toposes (or (∞,1)-toposes).” Some candidates are:
A geometric morphism that induces an equivalence of the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos, or more generally of shapes.
A geometric morphism which induces an isomorphism on all nonabelian cohomology with coefficients in constant ∞-stacks.
A geometric morphism which induces an isomorphism on all abelian sheaf cohomology with coefficients in locally constant sheaves (of complexes) of abelian groups, as well as nonabelian cohomology in dimension one (the most classical notion).
A geometric morphism which has an inverse up to “homotopy” in the classical sense of a map $E\times [0,1]\to F$, where since $E$ and $F$ are toposes, we have to incarnate $[0,1]$ as the topos $Sh([0,1])$.
Of these, the first three are arguably a notion of weak homotopy equivalence.
The first two definitions are equivalent, since (viewing a topos $E$ equivalently as the $(\infty,1)$-topos of sheaves on the same site, if necessary), a constant ∞-stack on $E$ is one of the form $LConst A$ for an ∞-groupoid $A$, and the cohomology of $E$ with coefficients in $LConst A$ is just $\pi_0(Hom_E(*,LConst A))$. But when $E$ is locally $\infty$-connected, $\Pi_E$ is left adjoint to $LConst$, so this is the same as $\pi_0 Hom_{\infty Gpd}(\Pi_E(*),A) = Hom_{Ho(\infty Gpd)}(\Pi_\infty(E),A)$ by definition of $\Pi_\infty(E)$. Thus, by the Yoneda lemma, a map of toposes induces an equivalence of fundamental ∞-groupoids, i.e. an isomorphism in $Ho(\infty Gpd)$, iff it induces an isomorphism on all such cohomology. Since in the non–locally-$\infty$-connected case, the shape is just the functor that would be represented by $\Pi_\infty(E)$ if it existed, the equivalence is even easier in that case.
These first two definitions also imply the third, since the abelian cohomology $H^n(E,A)$ with coefficients in an abelian sheaf $A$ can be identified with nonabelian cohomology in the locally constant stack $B^n A$ which is the $n$-fold delooping of $A$, and locally constant stacks on $E$ are represented by maps out of $\Pi_\infty(E)$.
Mike Shulman: There’s something a little funny here about “constant” versus “locally constant,” but I don’t have time to figure it out now. Maybe someone else can explain it.
Conversely, it is proven in Artin-Mazur that (in our language) a geometric morphism of 1-topoi satisfying the third definition necessarily induces isomorphisms on all homotopy progroups. (The idea is essentially that nonabelian cohomology is controlled by abelian cohomology with local coefficients, via Postnikov decompositions.) In the locally $\infty$-connected case, this is equivalent to inducing an equivalence of fundamental $\infty$-groupoids—the first definition. Thus, at least for locally $\infty$-connected 1-topoi, the first three definitions are all equivalent.
Mike Shulman: Possibly in the non–locally-$\infty$-connected case, the third definition is weaker, since a map of pro-$\infty$-groupoids can induce an isomorphism on all homotopy progroups without being an equivalence? It’s also not immediately obvious whether the result is also true as stated for all $(\infty,1)$-toposes.
I also seem to recall that the last notion of a “paths homotopy equivalence” implies the first few, but I don’t remember why.
One important thing to note is that any adjunction in the 2-category Topos induces a homotopy equivalence in any of the above senses. For the first (hence also the second and third) sense, we just observe that $\Pi_\infty$ is 2-functorial, whereas it lands in the (∞,1)-category ∞Gpd where all 2-morphisms are invertible; hence adjunctions get sent to adjunctions, which are necessarily equivalences.
For the final definition, we note that has tensors with the interval category, that are given by cartesian product with sheaves on the Sierpinski space $\Sigma$. Thus, a natural transformation between two geometric morphisms $E\to F$ is the same as a single geometric morphism $E\times Sh(\Sigma)\to F$. Now just pick a map $[0,1]\to \Sigma$ that doesn’t identify the endpoints, like the classifying map of $[0,\frac1 2]$, to get an actual “homotopy” between the same two geometric morphisms. Hence, any 2-morphism in $Topos$ gives a homotopy, so any adjunction gives a homotopy equivalence.
In particular, this means that any local geometric morphism or totally connected geometric morphism is a homotopy equivalence.
homotopy equivalence of toposes