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