I mailed in both 2019 and 2020 tax returns, but I still receive email notifications that my 2020 submission is rejected via Turbo Tax. Should I be concerned?

The software program does not know you mailed your returns.    If you know that you mailed them you can ignore the notices about filing.

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.

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s. 

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.

When the IRS opens its mail and begins to process mailed returns you can check the status on the IRS site. https://www.irs.gov/refunds

State returns have to be mailed to the state.

https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund

If you already mailed in the return then ignore the email alerts.  The TT program has no idea you mailed it in and the alerts will stop shortly.