Chances are that all the terms of the original lease survive, except those that are specific to the term of the lease. So if either party wants to stop renting, they could do so. But the monthly rent would be the monthly lease amount and if the lease had provisions, like no pets, that would continue.
It really depends on a number of things. You should contact an attorney to read the lease and advise you after collecting more facts.
Once the tenant has paid rent after the termination of the lease, a month to month tenancy is created. All of the terms and conditions of the expired lease, other than the term itself, carry forward into the month to month tenancy.
New York law provides that on the expiration of a written lease for a specific term (this does not apply to rent stabilization as the landlord must offer a renewal lease and the tenant must accept to avoid the risk of eviction) the subsequent month to month tenancy, that comes around after the landlord accepts rent for a month following the expiration of the lease, is on the same terms and conditions as the lease that expired, unless the landlord and the tenant agree in writing to different terms of the tenancy in the month to month period.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.