Big Sugar is being sued for health problems that environmental advocates have blamed on the annual burning of Florida’s sugar cane fields.

The would-be class action was filed June 4 on behalf of over 40,000 residents of the impoverished, rural communities squeezed between the fields and Lake Okeechobee. It asks a judge in U.S. District Court for the Southern District of Florida to block future pre-harvest fires in the fields.

It also seeks economic damages from the sugar cane producers and a court-supervised health monitoring program.

“Shutting them down is not the goal. The goal is full...