Are ammonia absorption tanks required on industrial refrigeration systems in Tennessee?

Yes, an absorption tank is likely to be required since the requirement is part of the current International Fire Code, which most jurisdictions in Tennessee have adopted. There are some exceptions for the use of a propane flare, or if you are under 22 lbs ammonia, or if a proper environmental study showing that environmental discharge will not result in a fire, health, or environmental hazard is accepted by the fire code official.

If the local jurisdiction has omitted the requirement from their local code (which they have the authority to do), then you are free to discharge into the atmosphere.

For your information, Section 606 of the IFC deals with mechanical refrigeration, and the absorption tank requirement and exceptions are found there.