So says the Tennessee Attorney General [1]:

An opinion by Tennessee Attorney General Robert E. Cooper Jr. says that making developers pay for new schools would be a development tax and is not allowed under existing law.

"(S)uch a provision would conflict with both the spirit and the express language of the County Powers Relief Act," the opinion states.

That act prohibits Tennessee counties from enacting new infrastructure taxes or fees, other than an adequate facilities tax in which the county can charge developers up to $1 per square foot for new construction. If a county enacts that tax, it has to eliminate all of its other infrastructure taxes.

[..]

Rutherford County has tried multiple times to get around the County Powers Relief Act, first by asking the attorney general if the county could increase its existing $1,500 per home development tax, then by asking Hood to send a request to Cooper's office to consider whether developers could be required to pay for new schools.

UPDATE: More from the Tennessean [2].

UPDATE: TN AG Nov. 21st opinion here [3].