Regardless of whether you e-file or file by mail, the due date for filing with an extension is October 15. If you file after that date you are filing late, whether you e-file or file by mail. If you are getting a refund, there is no penalty for filing late. If you have a payment due on a late-filed tax return you will owe a penalty for late filing.
Payment was due May 17 (for 2020 tax returns) even if you have an extension. The extension is an extension of time to file your tax return, not an extension of time to pay the tax. If you pay after May 17, you will owe a penalty and interest for late payment, even if you file by the October 15 extended due date.
TurboTax allows e-filing until October 15, whether or not you have an extension. It does not allow e-filing after October 15.
