NEW DELHI—A severe heat wave hanging over India has claimed more than 1,400 lives so far, and forecasters expect the high temperatures to continue several more days.

The states of Andhra Pradesh and Telangana in south India, where most of the deaths have occurred, got a respite Thursday with some light showers, according to B.P. Yadav, director of the India Meteorological Department in New Delhi. Still, the heat-wave warning has been extended three more days, he said, and high temperatures will “continue for at least one more...