Heavener is a town in Le Flore County, Oklahoma, United States. The population was 3,414 at the 2010 census, an increase of 6.7 percent …Heavener is a town in Le Flore County, Oklahoma, United States. The population was 3,414 at the 2010 census, an increase of 6.7 percent from 3,201 at the 2000 census. Heavener is notable for the Heavener Runestone just outside the city limits.