Sorafenib Tosylate in Treating Patients With Metastatic, Locally Advanced, or Recurrent Medullary Thyroid Cancer

Inclusion Criteria: * ELIGIBILITY CRITERIA SPECIFIC FOR ARM A * Histologically confirmed medullary thyroid carcinoma under the clinical setting of inherited tumor syndromes, such as multiple endocrine neoplasia (MEN) 2A, MEN 2B, or familial medullary thyroid carcinoma (FMTC) * ELIGIBILITY CRITERIA SPECIFIC FOR ARM B * Histologically confirmed medullary thyroid carcinoma under the clinical setting of sporadic medullary thyroid carcinoma (MTC) * ELIGIBILITY CRITERIA COMMON FOR ARMS A AND B * Patients must have measurable disease * Metastatic and/or locally advanced or locally recurrent disease * Oral or intravenous (IV) bisphosphonates therapy will be allowed for patients with bony metastasis at the investigator's discretion; bisphosphonate usage should be recorded if used since these agents may have anti-farnesyl transferase activity and may have some therapeutic effect in combination with sorafenib * Life expectancy must be \>= six months * Patients must have an Eastern Cooperative Oncology Group performance status 0-2 * Leukocytes \>= 2,000/uL (10 days prior to patient enrollment) * Absolute neutrophil count \>= 1,000/uL (10 days prior to patient enrollment) * Platelets \>= 100,000/uL (10 days prior to patient enrollment) * Total bilirubin =\< within 2 x upper limit of normal (10 days prior to patient enrollment) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\…