Florida has no personal income tax, so you will not pay tax to Florida.
New York has a "convenience" rule. What that means is that if you are telecommuting to the New York office (i.e. working remotely) because you want to live in Florida, you are still considered a New York employee. You are working in Florida for your own convenience. Your employer will continue to withhold New York state tax (but not city). You will have to file a New York nonresident tax return and pay New York state tax (but not city tax). However, if you are working in Florida because your employer wants to have an employee in Florida for business reasons, then you are there for the employer's convenience. You would be considered a Florida employee, and you would not pay any New York tax.
It doesn't matter where the business is incorporated.
