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.