I received a warrant 2 weeks ago from NYS IRS claiming I owe them money. About how long do I have to pay this before they execute a garnishment?

I'd recommend calling them, and trying to set up an agreed payment plan.  Having something you can give them on the account right now is usually a big help when talking about this.

I'm not a New York attorney, but it's my understanding that you would get a series of notices, and the first one gives you a deadline for contesting the debt (you're entitled to a hearing, if you disagree with the debt or the amount, and that first notice tells you how to ask for the hearing).  It's only after that, if you don't respond, that the actual warrant is issued, and that's as good as a judgment.  It can be executed upon at any time once it's served on you.