This randomized phase II trial studies how well pemetrexed disodium with or without crizotinib works in treating patients with stage IV non-small cell lung cancer that has progressed after crizotinib. Drugs used in chemotherapy, such as pemetrexed disodium, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Crizotinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether giving pemetrexed disodium is more effective with or without crizotinib in treating patients with non-small cell lung cancer that has progressed after crizotinib.