I have lived in several homes with
wood heat, one with propane. I haven't had an insurance company do anything other than send someone out to inspect the stove - but I have heard of some requesting a copy of the permit info for the stove.
A
rocket mass heater can be permitted in some jurisdictions as a masonry heater. I'm in an area still using 2012 IRC as the building code. Here is the Chapter that applies to masonry stoves:
Chapter 21 of 2012 IRC -Scroll down to section 2112 for Masonry Heaters
Please note that you need to scroll down on this page to get to the Masonry Heaters section.
In that section, it then says that a masonry heater will be installed either according to the manufacturer's guidelines, or using ASTM E 1602, which can be bought here:
ASTM E 1602 - Standard Guide for Construction of Solid Fuel Burning Masonry Heaters
Technically, people building masonry heaters are supposed to be experienced - or
guided by an experienced installer. I think this is where if someone wants to attempt building an RMH and having it be successfully permitted, getting consulting from one of the awesome RMH builders (who have dedicated their lives to this work in a most amazing way), could be a path to success. Nowadays, that could be done virtually.