It's been years but we've screenprinted a foam board called Artcore I think. It had a very thin plastic surface and we did a prototype run for Mott's apple juice which was shaped as an apple tree of course.
This was die cut after printing and had a beautiful soft pollowed edge after die cutting. I'm not sure that regular foamcore would cut as well without cracking or tearing the surface. I imagine it would depend on your shape.
If you're producing a rectangle, just run a hand roller over the egdes after cutting. I do that allot just because it gives it a nicer look than hard square cut edges.