A dozen communities in NH don't have property taxes. The students that live there are either homeschooled, unschooled or go to private schools. As for other government services, those are paid mostly by either user fees or the commercial timber operation tax.
As for alternatives to local property taxes paying for schools, we have two other options in NH. There is an online school paid for with state tax money, much of it from the lottery. It is free for NH residents. There are also state charter schools. They are paid by state taxes and the federal government so no property taxes go to them. They are also free to residents.
Send all kids to chartered schools or private schools and you can eliminate roperty taxes is a bunch more communities in NH and save the state a ton of money. Governments spend around $16000 a year per student on regular government schools but only around $6000 on chartered government schools.