No, merely owing New York State taxes is not a reason that the New York tax return cannot be e-filed.
More information may help resolve your issue, are you able to file your federal tax return? If your federal tax return cannot be e-filed, TurboTax cannot e-file your state tax return.
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This just happened to me. I finished my return and it was telling me that I had to mail in my NYS return because of a special situation but wouldn't tell me what. Go through your state return and delete all fields where you input 0. Then, if it still gives you the same issue, go to tax home, tools, delete forms, and delete all the NYS forms it allows you to, then reprocess it. It'll jump that you have to review a few fields - I had to revise my school district name, code and location, and once it reviewed that I was able to efile. It could be a different field within the NYS forms that's causing an issue but ultimately you will have to delete all the NYS forms, let it run again and find the error for you to correct. Hope this helps.