How bad would it be for a NiMH battery or/compared to a Lithium-ion battery in a device to sit unused for years?
With NiMH the shelf life is basically forever, the cells themselves will lose their charge in a few weeks or months, but after being stored they can be recharged without problems. There us no wear mechanism caused by storage.

Li-ion is different, if that runs flat it gets damaged (and cannot be recharged). 
But if you keep it topped up it can be stored for a long time too, and it retains its charge much better thabn NiMH.

Can a NiMH battery destroy a device?  I've never seen that happen, but I have seen a NiCd battery leak and damage the device. but this was during use, not during storage.