Few workers at the recycling plant had access to the secure room that some called the “Apple cage.” Behind its locked door, past a metal detector and under surveillance cameras, a small team of employees of GEEP Canada Inc., an electronic-waste processor north of Toronto, sifted through pallet-size boxes full of used iPhones. Prying each one open manually at a set of tables, they ripped out batteries and other parts and tossed the components into sorting bins. When enough material piled up in on...