Yes, they can raise the rent. However, the tenant has 30 days to 'decide' if they 'want' to stay and then another 30 days to vacate if that's what they choose.

Which is fair, depending on the increase. There is no limit in NH to how much a landlord can increase the rent, so this can be used as a method to evict a tenant who follows the rules but asks the landlord to make repairs and such.