I am familiar with Barnette's Lower Bound Theorem on the number of facets a $d$-dimensional simplicial convex polytope with $n$ vertices can have. Is there a similar result for a general (i.e. not necessarily simplicial) $d$-polytope with $n$ vertices?

A lower bound on the number of facets can be obtained from McMullen's upper bound theorem. It says that, for a given number of vertices, neighborly polytopes maximize the number of faces in all dimensions (among all polytopes, simplicial or not). In particular, a $d$-polytope $P$ with $m$ vertices has at most $F(d,m)$ facets (an exact formula for $F(d,m)$ can be found in Ziegler's "Lectures on polytopes"). By going to the dual polytope $P^*$ one sees that if the number $n$ of vertices of a $d$-polytope satisfies $F(d,m) \ge n > F(d,m-1)$, then it has at least $m$ facets.

For $n = F(d,m)$ this bound is exact (attained by the duals of neighborly polytopes). I do not know if for any $n$ inbetween there is a polytope with $m$ facets.