Abstract simplicial complexes happen to be posets, and to every abstract simplicial complex, one may associate a topological space, its geometric realization. This is as functor $\mathrm{Pos} \rightarrow \mathrm{Top}$. (asc's are defined by the property that if $A \in \Delta$ and $B\subseteq A$, then $B \in \Delta$)

Also, you might want to take a look into the theory of Bruhat--Tits buildings. Basically, one associates to a simple algebraic group a certain simplicial complex $\Delta(G)$. However, I have one tried to figure out if that association is a functor without any success since given up.

Both example's seem to be more geometric/topological/algebraic than really category theoretical (viz. more focused on the objects than the functor), but maybe still instructive.