Emergency officials in Rutherford County have started evacuating people downstream from Lake Lure and are sounding emergency sirens.
Water has been periodically released from the dam this week, but there has been too much rain to keep up, officials said.
Authorities did not indicate if the dam itself was in danger.
Lake Lure was transformed into upstate New York's Catskill Mountains for several of the scenes in the 1987 film, including the dance scene where actor Patrick Swayze lifts Jennifer Grey out of the water.
The Associated Press contributed to this report.