A taxpayer who becomes domiciled in a foreign country would only owe state income tax if they had income sourced to that particular state.
For example, a taxpayer who makes a permanent move to Canada, but who owns rental property in New York State, would still owe New York State income tax on their net rental income. Changing their mailing address to a PO Box in Florida or Texas would have no effect on their income tax obligation to New York.
I might add that the IRS does not accept a PO Box mailing address on a tax return unless "your post office doesn't deliver mail to your home." See page 14 of the Instructions for Form 1040:
https://www.irs.gov/pub/irs-pdf/i1040gi.pdf
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
