Unless your matrices have a special structure, both $A^{-1}$ and $A^{-1}B$ are dense (full), general matrices without any special properties. Even storing them in memory will be prohibitive.

The short answer is: don't do it. :)

The long answer is: look for structure in your matrices, and exploit it. If there is no structure and you really need to perform this computation, then your best bet is using an exact solver, not an iterative one. It will be extremely slow ($O(N^3)$ time and $O(N^2)$ storage), but there are no shortcuts.