A comonad is a triple $(F,\mu, \eta)$.
$$F: C \rightarrow C$$
F is an endofunctor on some category $C$.
$\mu$ is a natural transformation such that,
$$\mu: F \rightarrow F \cdot F$$
$\eta$ is a natural transformation such that,
$$\eta : F \rightarrow 1_C$$
Where $1_C$ is the identity functor on $C$.
I am interested in any monad where the functor $F$ maps an object to some well defined partial order on that object. The reason I am saying object instead of Set is that, while I am interested in comonads on Set, I am open to other ordered structures if they don't happen to be on Set. An example could be a functor that takes a set to the set of all partial orders on that set.
Do such comonads exist?
Any adjunction $G \dashv H$ induces a comonad $F=HG$ on the target of $H$.
You can find adjunctions where $H$ takes values in a category whose objects are partial orders. Does that work for you?
For example there is an adjunction $G\dashv H$ between $$G=(L\mapsto {\rm{Sp}} L):\rm{Loc}\to\rm{Top}$$ (the functor taking a locale to its space of points) and $$H=(X\mapsto \mathcal O_X):\rm{Top}\to\rm{Loc}$$ (the functor taking a topological space to its frame of open sets) where $\rm Top$ is the category of topological spaces and continuous functions and $\rm{Loc}$ is the category of locales, whose objects are complete Heyting algebras and whose morphisms are $\bigwedge$-preserving monotone functions with a $\wedge$-preserving left adjoint.
Composing the functors, you get a comonad $F=HG:\rm{Loc}\to Loc$.