Power companies that generate energy from decades-old waste piles they clean up would get a lifeline under a bill House Republicans have started moving forward.

Without the help, four of the nation’s 18 coal refuse plants—three in Pennsylvania and one in West Virginia—will likely go out of business in April 2019, said Sean Lane, executive vice president of governmental affairs at investment and management firm Olympus Power LLC. That’s when an existing waiver for waste-to-power companies under the EPA’s Mercury and Air Toxics Standards rule will expire.

The Satisfying Energy Needs and Saving the Environment (SENSE) Act (H.R. 1119...