TN doesn't have a regular income tax. Tennessee does have a state tax on interest and dividend income from investments. It is the only state tax that is levied on personal income. So, your wages are not subject to state tax.
The general rule, on state income tax, is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state.
In your case, neither state has an income tax. So you just don't have to file
