This phase I trial is studying the side effects and the best dose of MEK Inhibitor AZD6244 when given together with cetuximab in patients with advanced or refractory solid tumors that cannot be removed by surgery. MEK inhibitor AZD6244 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Monoclonal antibodies, such as cetuximab, can block tumor growth in different ways. Some block the ability of tumor cells to grow and spread. Others find tumor cells and help kill them or carry tumor-killing substances to them. Giving MEK Inhibitor AZD6244 together with cetuximab may kill more tumor cells.