One of the axioms for $(\infty,1)$-topoi is that the topos is disjoint, meaning that we have the following pullback diagram $$ \begin{matrix} 0 & \rightarrow & A \\ \downarrow & & \downarrow \\ B & \rightarrow & A \coprod B \end{matrix} $$ What is a non-example of an $(\infty,1)$-topos where this fails?

A basic non-example is the $(\infty,1)$-category of based spaces. Take $A = B = S^1$, then the homotopy pullback of the two inlcusions $S^1 \to S^1 \vee S^1$ is disconnected. In fact, it consists of countably many contractible components.