Labor unions face five lawsuits seeking a refund of more than $150 million in fees paid by nonmembers, and more than a dozen similar class actions could drain untold millions more from their coffers.

Public sector unions suffered a major financial setback in June, when the U.S. Supreme Court ruled in Janus v. AFSCME that they can no longer require nonmembers to pay fees for nonpolitical expenses, commonly known as “agency fees.” The new round of lawsuits seeks to recover...