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 similar chemical or biologic composition to sorafenib * No ongoing or active infection * No psychiatric illness or social situation that would preclude study compliance * No other uncontrolled illness * No other concurrent malignancy except nonmetastatic nonmelanoma skin cancer or carcinoma in situ of the cervix * No prior systemic chemotherapy for thyroid cancer (stratum I) * Prior systemic chemotherapy used to treat a second primary cancer with curative intent allowed provided the primary cancer was treated more than 5 years before study entry * No more than 3 prior systemic chemotherapy regimens for thyroid cancer (stratum II) * More than 6 weeks since prior systemic chemotherapy (stratum II) * No prior external beam radiotherapy to the sole site of measurable disease (except for patients with anaplastic thyroid cancer) * More than 6 weeks since prior external beam radiotherapy * More than 24 weeks since prior iodine I 131 * Recovered from all prior therapy * No prior sorafenib * More than 6 weeks since prior investigational tumor-specific therapy * Concurrent oral or IV bisphosphonates for bony metastases allowed at the discretion of the investigator * No other concurrent tumor-specific or investigational therapy * No concurrent combination antiretroviral therapy for HIV-positive patients * No concurrent therapeutic anticoagulation * Concurrent prophylactic anticoagulation (e.g., low-dose warfarin) for venous or arterial access devices allowed provided PT, INR, or PTT are normal