Rojo --
We have various water conservation (i.e., restricted water usage) programs in Tennessee, but for the most part these are under the auspicies of local governments and/or neighborhood associations, etc.
Right now there are various scattered communities throughout the area of middle TN that I know of that have imposed restrictions on water usage. So far, Nashville hasn't imposed any....I guess because the city is located right along the Cumberland River (even though the river itself is a lot lower than normal not only because of the drought, but also TVA and/or the Army Corps of Engineers has lowered at least one of the dams near the river's source in KY due to damage to that dam).
Actually the drought is worse the farther south and east you go from Nashville (towards, say, the city of Chattanooga).
Today's temperature in Nashville was predicted to hit 94 degrees F, thus making it the hottest day so far this year according to the official NWS records. Annual rainfall deficit for Nashville is now probably over 10 inches, and it's probably at least 15+ inches for areas in and around Chattanooga.
The state Department of Agriculture has requested that people do not make any outdoor fires or engage in any activities that could cause a fire to break out, but that is not a mandatory thing yet. Of course local governments or other local entities most likely have imposed such "no burning" regulations in areas under their juristiction. (You're not, for example, supposed to have any outdoor fires within Metro Nashville-Davidson County in the first place they tell me.)
The next rain for the Nashville area may occur on Tuesday, June 19, but only time will tell how true that may be.