Stick around for a response from any of the several experienced New York probate attorneys, but I believe that you're going to find that the answer is largely universal regardless of the jurisdiction. Until such time as the Estate no longer owns the property, the Estate is liable for the expenses of administration, including the upkeep and preservation of Estate assets. When the property is actually transferred to a beneficiary, the beneficiary accepts that property subject to indebtedness and ongoing expenses, absent some contradicting language in the Decedent's Will. Bottom line: It's likely the Estate's problem until it's actually transferred to the beneficiary.
This answer does not constitute legal advice. I am admitted to practice law in the State of Texas only, and make no attempt to opine on matters of law that are not relevant to Texas. This answer is based on general principles of law that may or may not relate to your specific situation, and is for promotional purposes only. You should never rely on this answer alone and nothing in these communications creates an attorney-client relationship.
Mr. Thomas is correct. The estate is responsible for the maintenance fee.
However, what I normally do is to quickly file application for "temporary letters" for the executor. This application allows the executor to receive "permission" from the Surrogate's Court to transfer the cooperative pending probate. This is generally an expedited application. The application allows the estate to transfer the asset quickly so as not to waste money.
I am a New york lawyer. You can apply for preliminary letters and get permission to use estate assets to pay the maintenance. However, this is only a safe bet if no one is contesting the will and the beneficiary is the same. Otherwise, you may be using estate assets that are meant for one beneficiary to pay expenses of a beneficiary of a sepecific bequest. Also, remember, co-operative shares are personal property.