Are there going to be any taxes due the state of NY upon the inheritance of my family homestead?

If you are referring to inheritance taxes, there are none in New York State. If you are referring to back property taxes, then you would have to ask your town hall, or whoever issues property taxes.

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It's possible that estate taxes may be due - but they would be paid out of the decedant's estate assets before the remaining assets are distributed to the heirs. The executor of the estate is responsible for determing whether any estate taxes are due, and if so he takes care of them. You as the heir have no tax liability due to your inheritance.