Who is responsible for property taxes when parent is in NH?

Technically it is the responsibility of the property owner to pay all taxes.
If rent is being paid in theory the amount of rent charged SHOULD cover taxes, mortgage if any and any utilities that are not changed to the renters name.

Since this is a family thing.....My thoughts here...
If brother is paying a fair market value for the rent then is the rent enough to cover taxes? If not and he is getting a "deal" he should step up and pay the property taxes.
If he was staying there and was a caregiver and rent was reduced to compensate for that then your Moms estate (her account) should pay for the property taxes.
Another opinion though...if everyone is getting along the property taxes should be paid through Mom's account. When the property is sold and the money placed into her account it will either be used to pay for her continuing care or it will eventually be divided between heirs as instructed by her Will.

If your brother lived in the home for 14 years taking care of your mom, why not take advantage of the child caregiver exemption and transfer the home from your mom to your brother. This will clarify the tax situation and keep the house out of MERP.

Your brother should be responsible.

Yes she is.

if your brother is living in the home, he should be responsible for property taxes. Is your mom on Medicaid?