I do not know where you live (original poster). It depends on the country, state, region, county and city that you live in. When building a new house in most of the western states (Rockies west) all inspections will all be done before the final is given, and any special permits or plan changes requiring permits will need to be done before that time (for everything, not just the stove). The general contractor is a licensed professional... he may or may not have to hire professinal and/or licensed sub-contractors, depending on state requirements. In Oregon a licensed electrician and plumber have to do electrical and plumbing work if the home is being sold or rented, but otherwise a general contractor or homeowner can do their own work and then have it inspected (even on plumbing and electrical). In Oregon, wood stoves need a permit and they have to be EPA certified, but anyone can install them as long as they meet the code and are inspected by the local county. In most of the western US, as a homeowner you can be your own contractor and pull permits and do your own work. I know that in the south and other places in the US, you cannot do that. Or if you do your own work, the inspection fees are the same as if they did the work themselves, specifically in states where contractors do their own inspections. Also there are city and state and county regulations regarding wood stove. In the SF Bay area, wood burning appliances cannot be installed in any new construction, period. In WA state all stoves have to meet strict WA state air quality standards and in many counties there they all have to have an OAK installed. Mobile/manufactured homes have their own requirements, and wood stoves have to be HUD approved to be installed in MHs anywhere in the US with an OAK. Many cities and regions have specific codes and variations on wood stove requirements, so you need to check with your city, county, and state before buying and installing a wood stove.
My insurance company gave me home insurance coverage with the wood stove, they never came by the house, and they did not raise any eyebrows in the process. Depends on the region and the insurance company I guess. My house in CA had a fireplace and that insurance company only came by to measure the distance from the nearest fire hydrant and fire station. Then did not bother to look inside.