Encorafenib and Binimetinib With or Without Nivolumab in Treating Patients With Metastatic Radioiodine Refractory BRAF V600 Mutant Thyroid Cancer

Inclusion Criteria:

* Ability to understand and the willingness to sign a written informed consent document
* Histologically (or cytologically) confirmed diagnosis of metastatic, radioiodine (RAI) refractory, BRAFV600E/M mutant differentiated thyroid cancer (DTC)

  * Note: RAI refractoriness is defined as:

    * The absence of uptake of RAI on either a low-dose diagnostic whole body scan, or a post-treatment RAI scan in measurable lesions
    * Radiographic progression of disease within 12 months of the last course of RAI treatment, or
    * Having a cumulative lifetime administered dose of \> 600 mCi of RAI
* Measurable disease meeting the following criteria and confirmed by radiography review:

  * At least 1 lesion of \>= 1.0 cm in the longest diameter for a non-lymph node or \> 1.5 cm in the short-axis diameter for a lymph node metastasis that is serially measurable according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 using computed tomography/magnetic resonance imaging (CT/MRI). If there is only 1 target lesion and it is a non-lymph node, it should have a longest diameter of \>= 1.5 cm
  * Lesions that have had external beam radiotherapy or locoregional therapies such as radiofrequency (RF) ablation must show evidence of progressive disease based on RECIST 1.1 to be deemed a target lesion.
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 1
* Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L
* Hemoglobin (Hgb) \>= 9 g/dL
* Platelets (PLT) \>= 75 x 10\^9/L
* Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) =\< 2.5 x upper limit of normal (ULN); participants with liver metastases =\< 5 x ULN
* Total bilirubin =\< 1.5 x ULN

  * Note: Individuals who have a total bilirubin level \> 1.5 x ULN will be allowed if their indirect bilirubin level is =\< 1.5 x ULN (i.e., participants with suspected or known diagnosis of Gilbert?s syndrome)
* Serum creatinine =\< 1.5 x ULN, or calculated creatinine clearance (determined as per Cockcroft-Gault) \>= 40 mL/min at screening
* Left ventricular ejection fraction (LVEF) \>= 50% as determined by a multigated acquisition (MUGA) scan or echocardiogram
* Triplicate average baseline corrected QT (QTc) interval =\< 480 ms
* Participants of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 72 hours prior to the start of assigned study intervention
* Participants of child-bearing potential agree to use highly effective methods of contraception starting with the first dose of assigned study intervention through 6 months after the last dose of study therapy
* Participants of childbearing potential are those who are not proven postmenopausal. Postmenopausal is defined as any of the following:

  * Amenorrheic for 1 year or more following cessation of exogenous hormonal treatments
  * Luteinizing hormone (LH) and follicle stimulating hormone (FSH) levels in the post-menopausal range for women under 50
  * Radiation-induced oophorectomy with last menses \> 1 year ago
  * Chemotherapy-induced menopause with \> 1 year interval since last menses
  * Surgical sterilization (bilateral oophorectomy or hysterectomy)
* Male participants must agree to use an adequate method of contraception starting with the first dose of study therapy through 6 months after the last dose of study therapy

Exclusion Criteria:

* Currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational agent/device within 4 weeks of first dose of study intervention

  * Note: Individuals in the follow-up phase of a prior investigational study may participate as long as it has been 4 weeks since last dose of the previous investigational agent of device
* Participants with active, known, or suspected autoimmune disease. Participants with type I diabetes mellitus, residual hypothyroidism due to autoimmune thyroiditis only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment are permitted to enroll
* Prior treatment with selective/potent BRAF/MEK inhibitors including vemurafenib, dabrafenib, encorafenib, selumetinib, trametinib, cobimetinib, binimetinib.

  * Note: Prior therapy with oral multikinase inhibitors (e.g., lenvatinib, sorafenib, cabozantinib, pazopanib, sunitinib, etc.) remain eligible for study participation.
* Prior treatment with anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CD137, or anti-CTLA-4 antibody (including ipilimumab or any other antibody or drug specifically targeting T cell co-stimulation or immune checkpoint pathways)
* Participants with a condition requiring systemic treatment with either corticosteroids (\>= 10 mg daily prednisone or equivalent) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids are permitted in the absence of active autoimmune disease
* History of allergy or hypersensitivity to any monoclonal antibody
* History or current evidence of retinal vein occlusion (RVO) or current risk factors to RVO (e.g. uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or hypercoagulability syndromes); history of retinal degenerative disease
* Previous or concurrent malignancy within 3 years of study entry, with the following exceptions:

  * Adequately treated basal or squamous cell skin cancer, superficial bladder cancer, prostate intraepithelial neoplasm, carcinoma in-situ of the cervix, or other noninvasive or indolent malignancy; OR
  * Other solid tumors treated curatively in which the expected rate of recurrence within 5 years is \< 5%
* Impaired cardiovascular function or clinically significant cardiovascular diseases, including any of the following:

  * History of acute coronary syndromes (including myocardial infarction, unstable angina, coronary artery bypass grafting, coronary angioplasty, or stenting) \< 6 months prior to screening
  * Symptomatic congestive heart failure (i.e. grade 2 or higher), history or current evidence of clinically significant cardiac arrhythmia and/or conduction abnormality \< 6 months prior to screening. Exceptions include asymptomatic/well-controlled atrial fibrillation/flutter or paroxysmal supraventricular tachycardia
  * Uncontrolled hypertension defined as persistent elevation of systolic blood pressure \>= 150 mmHg or diastolic blood pressure \>= 100 mmHg, despite medical therapy
* History of thromboembolic or cerebrovascular events =\< 12 weeks prior to the first dose of study treatment. Examples include transient ischemic attacks, cerebrovascular accidents, hemodynamically significant (i.e., massive or sub-massive) deep vein thrombosis or pulmonary emboli

  * Note: Individuals with either deep vein thrombosis or pulmonary emboli that does not result in hemodynamic instability are allowed to enroll as long as they are on a stable dose of anticoagulants for at least 4 weeks
  * Note: Individuals with thromboembolic events related to indwelling catheters or other procedures may be enrolled
* Known positive serology for HIV (human immunodeficiency virus), active hepatitis B, and/or active hepatitis C infection
* Known history of acute or chronic pancreatitis
* Impaired gastrointestinal function or disease that may significantly alter the absorption of encorafenib or binimetinib (e.g., uncontrolled vomiting or malabsorption syndrome)
* Concurrent neuromuscular disorder that is associated with the potential of elevated creatine kinase (CK) (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal muscular atrophy)
* Current use of a prohibited medication (including herbal medications, supplements, or foods), as described, or use of a prohibited medication =\< 1 week prior to the start of study treatment
* Any other condition that would, in the investigator?s judgment, contraindicate an individual?s participation in the clinical study due to safety concerns or compliance with clinical study procedures, e.g., infection/inflammation, intestinal obstruction, unable to swallow medication, social/ psychological issues, etc
* Participants who have undergone major surgery (e.g., intracranial, intrathoracic, or intra-abdominal surgery) =\< 3 weeks prior to starting study drug or who have not recovered from side effects of such procedure
* Participants that are pregnant or nursing (lactating)
* Prisoners or individuals who are involuntarily incarcerated
* Medical, psychiatric, cognitive or other conditions that may compromise the participant?s ability to understand the patient information, give informed consent, comply with the study protocol or complete the study