We recently hired an employee who has a residence in Florida, but is staying with relatives in NY while he works there. Do we deduct NY state & local income tax from him? Our co is based in PA, but he will only be working in NY.

Yes, you DO withhold at least NY state income taxes. All states are consistent in their position that ANY income earned within their state borders are subejct to their state income tax.

As for local income taxes, the duration of his work and the amount of money earned will determine whether he is also subject to the local income tax.

The employee will pay NY state income taxes since he/she will be earning while in NY. He/she will not pay PA taxes.
Read Your U.S. Tax Return: Working or Living in Two or More states