Iodine I-131 With or Without Selumetinib in Treating Patients With Recurrent or Metastatic Thyroid Cancer

Inclusion Criteria:

* Diagnosis of recurrent and/or metastatic thyroid cancer
* Histological or cytological confirmation of thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants); NOTE: medullary and anaplastic thyroid cancers are excluded; Hurthle cell carcinomas are excluded (defined as having an invasive tumor composed of \> 75% oncocytic \[Hurthle\] cells lacking the nuclear features of papillary carcinoma, tumor necrosis, and marked mitotic activity); patients with oncocytic (Hurthle cell) variants of papillary thyroid carcinoma (defined as a tumor composed of a majority of oncocytic \[Hurthle\] cells having the nuclear features of papillary carcinoma) are eligible to participate
* RAI-avid lesion on a radioiodine scan (a diagnostic, post-therapy, or post-ablation scans) performed =\< 24 months prior to registration, which suggests that therapy with 131I is justifiable in the judgment of the investigator
* Clinically or radiographically evident structural disease; patients with measurable disease and those with only non-measurable ("non-target") structural disease (according to modified Response Evaluation Criteria in Solid Tumors \[RECIST\] version \[v\] 1.1 criteria) are eligible;

NOTE 1: Modification of the RECIST v1.1 measurable disease criteria includes a change in the definition of what is considered a measurable malignant lymph node; a malignant lymph node is considered measurable if any of the following apply:

* It is noted to be RAI-avid on radioactive iodine imaging (diagnostic or post-therapy whole body scans acceptable) and it measures \>= 1 cm in the long axis,
* It is pathologically proven to be involved with thyroid cancer (by cytology or pathology) and it measures \>= 1 cm in the long axis, or
* Its short axis is \>= 1.5 cm when assessed by computed tomography (CT) scan NOTE 2: Patients only with biochemical evidence of disease without structural evidence of cancer are not eligible for this study

  * For patients with non-measurable, structural disease the following must apply:
* Undetectable thyroglobulin antibody AND
* A serum thyroglobulin of 10 ng/ml or greater in the context of suppressed thyroid-stimulating hormone (TSH) (TSH =\< 0.4 mcU/ml) =\< 28 days prior to study registration; use of any thyroglobulin assay is allowed, though all serum thyroglobulin measurements for study purposes must be conducted with the same thyroglobulin assay

  * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
  * Able to swallow and retain orally-administered medication with no clinically significant gastrointestinal abnormalities that may alter absorption
  * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 28 days prior to randomization)
  * Platelet count \>= 100,000/mm\^3 (obtained =\< 28 days prior to randomization)
  * Hemoglobin \> 9.0 g/dL (obtained =\< 28 days prior to randomization)
  * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 28 days prior to randomization)
  * Aspartate transaminase (AST) =\< 2.5 x ULN (or =\< 5 x ULN in presence of liver metastases) (obtained =\< 28 days prior to randomization)
  * Creatinine =\< 1.5 mg/dL OR calculated creatinine clearance of \>= 50 ml/min by either the Cockcroft-Gault formula or 24-hours urine collection analysis (obtained =\< 28 days prior to randomization)
  * Negative pregnancy test performed =\< 7 days prior to registration for women of childbearing potential only
  * Provide informed written consent
  * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)
* Note: during the Active Monitoring Phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up

  * Willing to provide mandatory archival tumor tissue (block or minimum of 30 unstained slides from a primary or recurrent/metastatic thyroid cancer) for correlative research purposes; NOTE: patients with less archival tumor tissue available may still be eligible for the study after discussion with Academic and Community Cancer Research United (ACCRU); receipt of archival tumor tissue is not required for study registration and initiation of therapy
  * Willing to provide mandatory blood samples for correlative research purposes

Exclusion Criteria:

* 131I therapy =\< 6 months prior to registration; Note: 131I administered solely for diagnostic purposes is not considered 131I therapy
* External beam radiation therapy =\< 28 days prior to registration; note: previous treatment with radiation is allowed if the investigator judges it will not compromise patient safety on the study
* Having been treated with a total cumulative (lifetime) 131I therapeutic activity \> 800 mCi (excluding 131I activity administered for diagnostic scans)
* Treatment with chemotherapy or targeted therapy (e.g. tyrosine kinase inhibitor) =\< 28 days prior to registration
* Prior exposure to mitogen-activated protein kinase kinase (MEK), RAS, or RAF inhibitors (note: previous exposure to sorafenib is allowed) OR history of hypersensitivity to selumetinib, thyrotropin alpha (Thyrogen), or any excipient agents
* Unresolved toxicity \> Common Terminology Criteria for Adverse Events (CTCAE) grade 2 from previous anti-cancer therapy, except for alopecia
* Cardiac conditions as follows:

  * Uncontrolled hypertension (blood pressure \[BP\] \>=150/95 mmHg despite medical therapy)
  * Left ventricular ejection fraction \< 55% measured by echocardiography
  * Atrial fibrillation with a ventricular rate \> 100 beats per minute (bpm) on electrocardiogram (ECG) at rest
  * Symptomatic heart failure (New York Heart Association \[NYHA\] grade II-IV)
  * Prior or current cardiomyopathy
  * Severe valvular heart disease
  * Uncontrolled angina (Canadian Cardiovascular Society grade II-IV despite medical therapy)
  * Acute coronary syndrome =\< 6 months prior to registration
* Ophthalmological conditions as follows:

  * Intra-ocular pressure \> 21 mmHg, or uncontrolled glaucoma (irrespective of intra-ocular pressure)
  * Current or past history of central serous retinopathy or retinal vein occlusion
* Symptomatic or untreated leptomeningeal disease, brain metastasis, or spinal cord compression
* Unable to follow a low iodine diet or requiring medication with high content in iodide (e.g., amiodarone)
* Received iodinated intravenous contrast within =\< 2 months of registration; avoidance of iodinated oral contrast is also preferred but not strictly required for study enrollment; NOTE: those who have had iodinated intravenous contrast within this time frame may still be eligible if a urinary iodine analysis reveals that excess iodine has been cleared (defined as urinary iodine documented to be \< 300 mcg/day by either a spot urinary iodine or 24-hour urinary iodine measurement)
* Any of the following:

  * Pregnant women
  * Nursing women
  * Men or women of childbearing potential who are unwilling to employ adequate contraception
* Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy; NOTE: patients known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, hepatitis B infection, hepatitis C infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm. (NOTE: Performance of investigational 124I PET/CT scans is allowed prior to and during conduct of this study)
* Other active malignancy =\< 2 years prior to registration that will interfere with conduct of this trial; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history of prior malignancy, patients must not be receiving other specific treatment (chemotherapy, hormonal therapy, radiation) for their cancer
* Not willing to discontinue use of supplemental vitamin E