This randomized phase II trial studies the effects, good and bad, of using everolimus along with sorafenib tosylate versus sorafenib tosylate alone in treating patients with advanced radioactive iodine refractory thyroid cancer. Sorafenib tosylate and everolimus may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. The addition of everolimus to sorafenib tosylate may cause more shrinkage of thyroid cancer and may prevent it from growing but it could also cause more side effects than sorafenib tosylate alone. It is not yet known whether this treatment with sorafenib tosylate and everolimus is better, the same, or worse than sorafenib tosylate alone.