If you're a resident of New York, ALL your income is taxable by New York, regardless of where you earned it. Therefore you must file a New York tax return.
Rhode Island can only tax a non-resident on income earned from work actually (physically) performed in Rhode Island. Therefore, if you never lived in, or physically worked in, Rhode Island, your income is not taxable by Rhode Island. Some states tax non-resident remote workers, but Rhode Island is not one of them. If this is your situation, and your employer mistakenly withheld RI taxes, you'd have to file a non-resident RI return, showing zero RI income, in order to have those taxes refunded to you. I suggest you file that return on paper, and attach a letter of explanation.
Finally, if your employer will not or cannot withhold NY taxes, you may want to start making quarterly estimated tax payments to NY.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
