NEW YORK STATE AGRICULTURE & MARKETS LAW

331. Abandonment of certain animals.

An animal is deemed to be abandoned when it is placed in the custody of a veterinarian, veterinary hospital, boarding kennel owner or operator, stable owner or operator, or any other person for treatment, board or care and:

1. Having been placed in such custody for a specified period of time the animal is not removed at the end of such specified period and a notice to remove the animal within ten days thereafter has been given to the person who placed the animal in such custody, by means of registered letter mailed to the last known address of such person; or
2. Having been placed in such custody for an unspecified period of time the animal is not removed within twenty days after notice to remove the animal has been given to the person who placed the animal in such custody, by means of a registered letter mailed to the last known address of such person.
3. The giving of notice as prescribed in this section shall be deemed a waiver of any lien on the animal for the treatment, board or care of the animal but shall not relieve the owner of the animal removed of his contractual liability for such treatment, board or care furnished.

332. Disposition.

Any person having in his care, custody or control any abandoned animal, as defined in section three hundred thirty-one of this chapter, may deliver such animal to any humane society or society for the prevention of cruelty to animals having facilities for the care and eventual disposition of such animals, or, in the case of dogs, cats and other small animals, to any pound maintained by or under contract or agreement with any county, city, town or village within which such animal was abandoned. The person with whom the animal was abandoned shall, however, on the day of divesting himself of possession thereof, notify the person who had placed such animal in his custody of the name and address of the animal society or pound to which the animal has been delivered, such notice to be by registered letter mailed to the last known address of the person intended to be so notified. If an animal is not claimed by its owner within five days after being so delivered to such humane society or society for the prevention of cruelty to animals, or pound, such animal may at any time thereafter be placed for adoption in a suitable home or euthanized. In no event, however, shall the use of a decompression chamber or decompression device of any kind be used for the purpose of destroying or disposing of such animal. (emphasis added)

NYSHA's notes on Article 25B, Sections 331 and 332
If an owner has placed an animal in one of the aforementioned facilities (custody of veterinarian, veterinary hospital, boarding kennel owner or operator, stable owner or operator, or any other person for treatment, board or care) for a specified period of time, the owner must remove the animal from the facility within ten days from the time that he receives a notice by means of registered mail.

If the animals was placed in one of the aforementioned facilities for an unspecified period of time, the owner has twenty days to remove the animal after being given notice by registered mail.

If the owner does not redeem his animal within the specified period of time, the owner/operator of the boarding facility may deliver the animal to any humane society or SPCA having facilities to care for the animals, or in the case of dogs or cats, to a facility within such town or village where the animal was abandoned. The owner/operator of the boarding facility shall then notify the animal's owner as to where the animal was placed.

Additional laws to consider:
See Article 26 Section 355, which deals with animals being abandoned in a public place by the animal's owner or other person in charge of the animal.