It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^*B' = A'$; when $i$ is mono, this is always possible, with canonical solutions given by the direct image $\exists_i A'$ and the dual image $\forall_i A'$.

Nothing here is special to simplicial sets: this argument anpplies equally in any model structure on a topos in which all cofibrations are monomorphisms.

I’m not certain of a precise reference for this in the literature.  I vaguely recall it appearing in Cisinski 2006, [*Les préfaisceaux comme modèles des types d’homotopie (Presheaves as models for homotopy types)*, 2006](http://www.numdam.org/item/AST_2006__308__R1_0/), or in one of his related papers, but on a quick skim now I can’t locate it.  A similar but slightly harder result — fibrancy of a universe classifying a certain class of fibrations, corresponding to showing that those fibrations extend along trivial cofibrations — appears in my 2012 paper with Chris Kapulkin, [*The simplicial model of univalent foundations (after Voevodsky)*](https://arxiv.org/abs/1211.2851), in Sections 2.1, 2.2, and 3.2.