Sculpted images intended for active worship were not painted, with some notable exceptions. Worshiped deities are frequently ritually bathed, and the traditional pigments are water soluble. The major exception to this rule are the wooden deities in the Jaggannath Temple in Puri, which are painted neem wood. These are bathed once a year, and are ritually repainted.

Ancient secular sculptures were likely not painted either. The Didarganj Yakshi (~ 3rd Century BCE) is a prime example, and demonstrates the high mirror sheen associated with the Mauryans. Other sculptures from the time period, such as the railings of the Sanchi Stupa built during the Emperor Ashoka's lifetime, show no discernible paint residue. Yet it is quite possible that artwork on the ancillary structures were painted, as they were donated by the lay public, as per their individual capacities.

Since OP's question references the sculptures on Gopurams, they were indeed likely painted, even in ancient times, with some exceptions. Unfortunately most older gopurams are destroyed, either by medieval Islamic armies, or collapsed in earthquakes. Most gopurams standing today were rebuilt in the last few hundred years. Some old gopurams were repainted with enamel paint decades ago, destroying evidence of the original paint residue.

We have strong reason to believe the old gopurams would have been painted, since it's instructed to be done so in the temple building manuals, the shilpashastras. The shilpashastras are a wide ranging series of scriptural texts likely composed sometime after the 8th Century CE. The texts deal with a wide array of topics, ranging from architecture, sculpture, metallurgy, carpentry, and painting. The Varnasamskara lakshanam of the shilpashastras deals with the painting of religious icons. These would have been painted according to the rules of the section on Varnalepna. These specify the use of five approved colours, the panchavarna (red, yellow, black, white, green).