For the lattices of all subsets of a given set, an axiomatic characterization is known: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The question: How to characterize the sets of filters on a set? That is, having a poset, how to check whether it is isomorphic to the set of filters on some set?
Note that we allow improper filters (An improper filter is also filter) to ensure that the set of filters is a complete lattice.
Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. Translating this idea to a purely point-free context, the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.
