"For example, the local property taxes on my single family residence is $8K. This is for a house that is currently worth about $340K."We're paying about $9K here in the People's Republic on a house that's maybe worth $400K (Zillow says $500K. They gotta be kiddin'!) My late FIL's place -- 10 x the size and closer to the water -- had a $40K tax bill and that was 10 years ago. The problem here in the People's Republic is that everything else - income, sales, inheritance, business -- is taxed to the max, too.
My thought is that I doubt that NH real estate would be hurt too much. In Massachusetts -- one of the Seven Deadly (tax) States, it would. It's the overall tax burden, not any one single tax source.
The high tax states are in trouble. Mostly what the tax bill would do is make them advertise that fact. But I don't think it's going to happen.
In NH property tax rates vary from town to town typically based on the amount of school age children in the town, how fancy your school system is and the amount of commercial real estate exists in your locality.
Tuftonboro, NH and Newington, NH are two towns with very low property tax rates. The first is rural community in the central part of the state. The later is in the seacoast area that has a large amount of commercial real estate along a river. If you reside in one of these towns you literally could have one of the cheapest local/state tax burdens in the country and still be within an 1-2 hour drive from Boston.