Sunitinib Malate in Treating Patients With Iodine-Refractory Recurrent or Metastatic Thyroid Cancer

Inclusion Criteria:

* Histologically or cytologically proven metastatic WDTC or MTC
* Evidence of refractoriness to iodine therapy for WDTC documented by a combination of imaging and thyroglobulin or by biopsy
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 3
* Evidence of fludeoxyglucose F 18 (FDG) PET avid metastatic tumors
* Measurable disease by RECIST criteria
* Resolution of all acute toxic effects of prior systemic therapy (including iodine therapy or chemotherapy), radiotherapy or surgical procedure to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) Version 3.0 grade =\< 1
* Total serum bilirubin =\< 1.5 x upper limit of normal (ULN) (patients with Gilbert's disease exempt)
* Serum transaminases =\< 2.5 x ULN or =\< 5.0 X ULN if secondary to liver metastases
* Serum creatinine =\< 1.5 x ULN
* Absolute neutrophil count (ANC) \>= 1.5 X 10\^9/L
* Platelets \>= 100,000/uL
* Hemoglobin \>= 9.0 g/dL
* Willingness and ability to comply with scheduled visits, treatment plans and laboratory tests and other study procedures
* Male and female patients with reproductive potential must use an acceptable contraceptive method
* Signed and dated informed consent document indicating that the patient has been informed of all the pertinent aspects of the trial prior to enrollment

Exclusion Criteria:

* Concomitant treatment in another therapeutic clinical trial
* ECOG performance status \>= 3
* Symptomatic, untreated, brain metastasis
* Second primary malignancy that is clinically detectable or clinically significant at the time of consideration for study enrollment
* Full-dose anticoagulation defined as:

  * Low molecular weight heparin use with the intent of full dose anticoagulation; example: enoxaparin 1.5 mg/kg daily or equivalent
  * Warfarin use to keep international normalized ratio (INR) greater than or equal to 2
* History of gross hemoptysis (defined as bright red blood of at least 1/2 teaspoon or 2.5 mL per episode) within 3 months prior to study drug administration unless definitively treated with surgery or radiation
* Any of the following within the 6 months prior to study drug administration: myocardial infarction, severe/unstable angina, coronary/peripheral artery bypass graft, symptomatic congestive heart failure, cerebrovascular accident or transient ischemic attack, or pulmonary embolism; ongoing cardiac dysrhythmias of NCI CTCAE Version 3.0 grade \>= 2
* Type I Diabetes Mellitus; patients with Type II Diabetes Mellitus will be included as long as their glucose can be controlled between levels of 80 and 150 mg/dL
* Uncontrolled Hypertension (\> 150/100 mm Hg despite optimal medical therapy)
* Major surgery or radiation therapy within 4 weeks of starting the study treatment
* Other severe acute or chronic medical or psychiatric condition, or laboratory abnormality that would impart, in the judgment of the investigator, excess risk associated with study participation or study drug administration, or which, in the judgment of the investigator, would make the patient inappropriate for entry into this study
* Pregnancy or breast feeding