The category of posets has no subobject classifier. Indeed, if there did exist a universal subobject $1\to \Omega$, then it would be a regular subobject, and so every subobject would be regular since regular subobjects are stable under pullback. But the regular subobjects are exactly the "subposets" in the usual sense (i.e., subsets with the inherited order), and so not all monomorphisms of posets are regular (because the domain can have a weaker order than the codomain).
[In fact, if you work with "evil" posets (i.e., antisymmetric preorders), there are even regular subobjects which cannot be pulled back from any terminal subobject (here by "terminal subobject" I mean a subobject $A\to B$ where $A$ is a terminal object, as a universal subobject $1\to\Omega$ would have to be). For instance, it is easy to see that if $P$ is a poset and $Q\subseteq P$ is the pullback of a terminal subobject $1\to R$ via some map $P\to R$, then for all $a,b\in Q$ and all $c\in P$, $a\leq c\leq b$ implies $c\in Q$.]