I buy pool table felt from a local company that builds and restores pool tables, and this felt wont ever cause rusting. I use Elmers construction glue to install dividers, and also the felt lining. Never had issues with rusting.
I have a felt lined custom case that came with my Pope three barrel set that's over 125 years old, and always had the gun, barrels, scope and all the tools in the case. Whatever felt they used back then must be great as it's the only rifle I leave inside a case for long periods.