Just check with your local tax assessor's office, or better yet ask someone who has a barn so you don't raise an eyebrow with your town. I think NY would tax the air if they could at this point, so I'd have to guess you would be assessed for the additional structure, unless you're way upstate in the middle of nowhere.
You have increased the value of your property so your assessment will go up.
In my town (also in NY) my assessment went up by an amount equal to what the barn builder put on the permit application regarding how much the barn was costing us.
We tried to get the assessment changed (because we all know that it does not matter how much the barn cost, it did not raise my property values by that amount) and they refused to lower the assessment.