Is it safe to operate a kitchen stove without the original insulation installed?
No. Manufacturers don't spend money doing things like installing insulation if they're not important. 

This is especially true where you have tenants who are less concerned with careful use of the appliances than you'd be, and where you are potentially liable for injury or death. 

If by "my rental house" you actually meant "the house I rent", you'd need permission from the owner (but don't expect to get it, for the reasons I mentioned).