Tennessee does not impose an income tax, except for a so-called "Hall" income tax on interest and dividends, so No, you do not need to file a Tennessee Income tax return; however, according to the Tennessee Department of Revenue:
If you are a corporation, limited partnership, limited liability company, or business trust chartered/organized in Tennessee or doing business in this state, then you must register for and pay franchise and excise taxes. The franchise tax is based on the greater of net worth or the book value of real or tangible personal property owned or used in Tennessee. The excise tax is based on net earnings or income for the tax year.
However, you should qualify for an exemption as a "LLCs or LPs where the members are fully-obligated for the debts of the entity (OME)"
For more information, see Tennessee Franchise & Excise Tax
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