I have some loaded ammo that is pretty badly tarnished. Can I just put it in my tumbler and clean it up?

No. Aside from the possibility of a sharply pointed bullet striking a primer and causing a detonation, the deterrent coating may be altered, which speeds the burning rate. Most tumblers would require the ammo to be tumbled for quite some time before it was thoroughly cleaned, possibly long enough to cause some breakdown of the powder or its deterrent coating. The simple answer here is, “No, don’t do it.”