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 \[SGPT\]) =\< within 3 x upper limit of normal (10 days prior to patient enrollment) * Serum creatinine within normal institutional limits OR creatinine clearance \> 30 mL/min (by Cockcroft-Gault formula) (10 days prior to patient enrollment) * The effects of sorafenib (BAY 43-9006) on the developing human fetus at the recommended therapeutic dose are unknown; for this reason and because kinase inhibitors are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 30 days after completion of therapy; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * EXCLUSION CRITERIA FOR ARM A AND B * Patients who have had systemic anti-tumor therapy (such as chemotherapy, biologic modifiers or antiangiogenic therapy) within 4 weeks (6 weeks if nitrosourea or mitomycin chemotherapy) prior to study entry * Patients who have had external beam radiation therapy within 1 week or if the adverse events associated with radiation are not resolved to grade 1 or less prior to study entry * Prior therapy with sorafenib (BAY 43-9006), ZD 6474 or AMG-706 * Patients currently receiving any other tumor-specific therapy for thyroid cancer or investigational therapy; patients receiving adjuvant hormonal therapy for a second primary (such as breast cancer or prostate cancer) are allowed to participate as far as there are no known drug interactions * History of allergic reactions attributed to compounds of similar chemical or biologic composition to sorafenib (BAY 43-9006) * Patients unable to swallow sorafenib tablets (e.g. any condition that impairs patient's ability to swallow pills) * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, uncontrolled hypertension, or psychiatric illness/social situations that would limit compliance with study requirements * Patients with any evidence of a bleeding diathesis * Patients actively receiving anticoagulation with therapeutic intent; prophylactic anticoagulation (i.e. low dose warfarin) or venous or arterial access devices is allowed provided that the prothrombin time (PT), international normalized ratio (INR) or partial thromboplastin time (PTT) are normal * Pregnant women or women who are breast-feeding are excluded from this study because sorafenib (BAY 43-9006) is an investigational agent and teratogenicity has not been evaluated yet; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with sorafenib (BAY 43-9006), breastfeeding should be discontinued if the mother is treated with sorafenib (BAY 43-9006) * Human immunodeficiency virus (HIV)-positive patients receiving combination anti-retroviral therapy because of possible pharmacokinetic interactions with sorafenib (BAY 43-9006); patients with immune deficiency are at increased risk of lethal infections when treated with marrow-suppressive therapy * Patients taking the cytochrome P450 enzyme-inducing antiepileptic drugs (phenytoin, carbamazepine, or phenobarbital), rifampin or St. John's wort due to potential drug interactions with sorafenib