TurboTax will not know anything about your mailed return, and will continue to show “Ready to Mail” on your account. TurboTax will not know that you put your tax return in an envelope and took it to a mailbox. TurboTax does not get updates on mailed (or e-filed) returns.
Wait about 4 weeks from when you mail a return and then you can check the status on the IRS site. https://www.irs.gov/refunds
State returns have to be mailed to the state.
If you mail a tax return (or a payment) to the IRS, it is a good idea to use a mailing service that will track it like UPS or certified mail so you will know it was received.