No. Tennessee does not impose an income tax, except for a so-called "Hall" income tax on interest and dividends:
You only need to file a return for the Tennessee Hall tax on interest and dividends, at a rate of 3%, if you are a person whose legal domicile is in Tennessee, and whose taxable interest and dividend income exceeded $1,250 ($2,500 if married filing jointly) during the tax year.
You must report any rental income on your federal tax return, of course.
[Edited 4/15/19 | 10:50am]
