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

Inclusion Criteria: * Histologically confirmed diagnosis of 1 of the following: * Papillary thyroid cancer (stratum I) * Papillary, follicular, Hurthle cell, insular, or anaplastic thyroid cancer (stratum II) * Mixed histology, poorly differentiated, or tall-cell variants allowed * Metastatic, locally advanced, or locally recurrent disease * At least 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan * The following are considered non-measurable disease: * Tumors in a previously irradiated area * Bone lesions * Leptomeningeal disease * Ascites * Pleural/pericardial effusion * Lymphangitis cutis/pulmonis * Abdominal masses not confirmed and followed by imaging techniques * Cystic lesions * Archival tumor tissue block OR material collected before study entry available (stratum I) * Biopsy-accessible disease (stratum I) * Performance status - ECOG 0-1 * At least 6 months * WBC \>= 3,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * No bleeding diathesis * Bilirubin =\< 1.5 times upper limit of normal (ULN) * AST and ALT =\< 1.5 times ULN * Creatinine =\< 1.5 times ULN * No uncontrolled hypertension * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * Willing to undergo 2 tumor biopsies during study participation (stratum I) * No history of allergic reaction attributed to compounds of sim…