If you live in a state that collects School District taxes, TurboTax will cover the topic in your state return. Even if you don't owe School District taxes, you'll still need to indicate which district you live since your state requires a district code.
You'll usually be asked to enter the code after identifying the name of the county you reside in, depending on your state. Often, you'll simply choose from a drop-down list in TurboTax (click image for example from Ohio's return)