The fast-food industry got the world hooked on cheap burgers, chicken nuggets and pizza. Now a $6.5 trillion investor coalition wants it to overhaul the supply chains that make those menus possible.

Led by sustainable development advocates FAIRR Initiative and Ceres, the group sent letters to Domino’s Pizza Inc., McDonald’s Corp., Burger King owner Restaurant Brands International Inc., Chipotle Mexican Grill Inc., Wendy’s Co. and Yum! Brands Inc. The letters outline the environmental impact of their meat and dairy supply chains and urge the companies to change.

Investors are asking the companies to respond by March 1 on plans to...