I'm assuming your question involves just *good* prime characteristic $p$.   Much of the literature focuses on *unipotent classes*, but Springer's $G$-equivariant isomorphism (for good $p$) between the unipotent variety in $G$ and the nilpotent variety in $\mathfrak{g}$ shows that the classes and orbits are in bijection and also allows one to transfer the closure relationships.   Thus the closure ordering graphs are the same for the group and the Lie algebra.   (Here one has to be a bit careful about the isogeny type of $G$ in type $A_n$, however.)

It turns out after some work that the closure orderings of unipotent classes are the same as those found much earlier by Gerstenhaber and others in characteristic 0.  Much of this work was done by Spaltenstein (and those he cites including Mizuno, Shoji for exceptional types): see his *Classes unipotentes et sous-groupes de Borel*, Lect. Notes in Math. 946 (Springer, 1982), especially II.8 and the graphs for exceptional types in IV.2.   [Carter's 1985 book follows this development, though for types $E_7, E_8$ on pages 442 and 444 the graphs lack several edges; this was probably a technical error made during the production of the book.]  Though I've never checked all the details carefully, I've been assured that experts have done so and find Spaltenstein's results convincing.