The withholding will be screwed up, but if you claim to be in NY and are actually in Wyoming, you'll probably get a big refund.

But you'll file in your real state of residence, and the other state will then insist you owe them taxes because your employer reported you were working there.

That could be an awkward conversation since a W-4 has this above your signature: "Under penalties of perjury, I declare that this certificate, to the best of my knowledge and belief, is true, correct, and complete."

Definitely consult a tax attorney before you try anything like this.

Fraud is what you're talking about doing and it is illegal. Also most states require payroll taxes so the company would be on the hook for the lie too. You'd be lucky if you only got fired.