Yes, if the sign meets the requirements of T.C.A. 39-17-1359 it carries theme weight if law.
Fallguy is correct. Here's some more specifics.
Signs carry the weight of law if they meet the requirements of at least one of two options. Option 1 is to have wording that is "substantially similar" to the following: "AS AUTHORIZED BY TCA § 39-17-1359, POSSESSION OF A WEAPON ON POSTED PROPERTY OR IN A POSTED BUILDING IS PROHIBITED AND IS A CRIMINAL OFFENSE." To be "substantially similar," the wording must state that "(i) The property is posted under authority of Tennessee law; (ii) Weapons or firearms are prohibited on the property, in the building, or on the portion of the property or building that is posted; and (iii) Possessing a weapon in an area that has been posted is a criminal offense." Option 2 is for the sign to have a firearm with "The international circle and slash symbolizing the prohibition of the item within the circle" over top of it. Carrying past such a sign is a class B misdemeanor, punishable by a fine of $500.
Now, there are a couple of possible loopholes. First, the law requires that signs must be "displayed in prominent locations, including all entrances primarily used by persons entering..." So, if a building has three entrances and only two are posted, the building isn't properly posted. Second, the signs must meet the requirements for one of the two options listed above. So, for example, the signs at many of the malls in Tennessee that say simply "Firearms and other weapons are not allowed" arguably do not meet the requirements for the law. I will note, however, that I do not have any plans to personally test any of these loopholes. Using them could easily make you a test case. Proceed at your own risk.
Disclaimer: I am not a lawyer. None of the above information is, or should be construed as, legal advice. I accept no criminal or civil responsibility or liability for any actions you may or may not take based on this information. For actual legal advice, consult an attorney that is licensed in your jurisdiction.