Rather obvious in retrospect, but all 01-polytopes are inscribed (the vertices lie on a sphere). So if $P$ is non-inscribable, then it can't be a 01-polytope. There are non-inscribable rational polytopes already in dimension three.
Another property of 01-polytopes not shared by all polytopes is that all 2-faces are 3-gons or 4-gons (this is not hard to see but probably proven somwhere in "Lectures on 01-polytopes" by Ziegler). So e.g. the pentagon is not a 01-polytope.
It would be interesting to see an inscribable rational polytope with only 3- and 4-gonal 2-faces that is still not a 01-polytope.