Lenvatinib and Pembrolizumab in Differentiated Thyroid Cancers (DTC)

Inclusion Criteria:

* Locally recurrent and unresectable and/or distant metastatic differentiated thyroid cancer (DTC), histologically or cytologically confirmed; the diagnosis of DTC includes the following subtypes: papillary thyroid cancer (PTC) (including but not limited to variants such as follicular variant, tall cell, columnar cell, Hurthle cell variant of papillary carcinoma, and poorly differentiated), follicular thyroid cancer (FTC), including insular variant, Hurthle cell carcinoma and poorly differentiated thyroid cancer
* Measurable disease meeting the following criteria:

  * 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 which is serially measurable according to RECIST 1.1 using computerized tomography/magnetic resonance imaging (CT/MRI); if there is only one 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 (EBRT) or loco-regional therapies such as radiofrequency (RF) ablation must show evidence of progressive disease based on RECIST 1.1 to be deemed a target lesion
* For cohort 1 only: evidence of disease progression =\< 14 months prior to registration according to RECIST 1.1, as confirmed by the site study principal investigator (PI)
* For cohort 2 only: progressive disease (PD) on lenvatinib per RECIST 1.1 =\< 60 days prior to registration, as confirmed by the site study PI; patients need to have documented imaging and measurement of RECIST target lesions within 30 days of starting pembrolizumab
* Radioiodine (RAI)-resistant disease as defined by one or more of the following criteria:

  * One or more measurable lesions that do not demonstrate RAI uptake
  * One or more measurable lesions progressive by RECIST 1.1 =\< 14 months of prior RAI therapy
  * One or more measurable lesions present after cumulative RAI dose of \>= 600 mCi
  * One or more measurable lesions that are fludeoxyglucose F-18 (FDG)-avid (\> 5 standardized uptake value \[SUV\]), if positron emission tomography (PET)/CT scan performed; these lesions may also be RAI-avid
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1
* Absolute neutrophil count (ANC) \>= 1,500 /mcL (obtained =\< 30 days prior to registration)
* Platelets \>= 100,000 / mcL (obtained =\< 30 days prior to registration)
* Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin dependency (=\< 7 days prior to registration) (obtained =\< 30 days prior to registration)
* Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN (obtained =\< 30 days prior to registration)
* Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN (obtained =\< 30 days prior to registration)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for subjects with liver metastases (obtained =\< 30 days prior to registration)
* Albumin \>= 2.5 mg/dL (obtained =\< 30 days prior to registration)
* International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants (obtained =\< 30 days prior to registration)
* Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants (obtained =\< 30 days prior to registration)
* Adequately controlled blood pressure with or without antihypertensive medications defined as blood pressure (BP) \< 150/90 mmHg at screening
* Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only
* Ability to complete patient medication and blood pressure diaries by themselves or with assistance
* Willing and able to 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 tissue and blood samples for correlative research purposes

Exclusion Criteria:

* Cohort 1 only: prior treatment with previous VEGFR active multikinase inhibitor
* Cohort 2 only: discontinued lenvatinib due to toxicity
* Pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment
* Female subjects of childbearing potential: unwilling or unable to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication; NOTE: subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year
* Male subjects: unwilling or unable to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy
* 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 (HIV 1/2 antibodies) and currently receiving antiretroviral therapy
* Currently participating and receiving study therapy (except lenvatinib for patients in cohort 2) or has participated in a study of an investigational agent and received study therapy within 4 weeks prior to registration
* Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy =\< 7 days prior to the first dose of trial treatment
* Known history of active TB (Bacillus tuberculosis)
* Hypersensitivity to pembrolizumab or any of its excipients
* Prior anti-cancer monoclonal antibody (mAb) =\< 4 weeks prior to registration or who has not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to agents administered \>= 4 weeks prior to registration
* Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study day 1 (except lenvatinib for patients in cohort 2) or who has not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to a previously administered agent

  * NOTE:

    * Subjects with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study
    * If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to registration, as deemed by treating investigator or site PI
* Known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer
* Known active central nervous system (CNS) metastases and/or carcinomatous meningitis; NOTE: subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for \>= 7 days prior to trial treatment; this exception does not include carcinomatous meningitis which is excluded regardless of clinical stability
* Active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs); NOTE: replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment
* Known history of, or any evidence of active, non-infectious pneumonitis that required steroids
* Active infection requiring systemic therapy
* History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator
* Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial
* Received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent
* Known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected)
* Received a live vaccine =\< 30 days of planned start of study therapy; NOTE: seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed
* Proteinuria \> 1+ on dipstick urinalysis; patients with \> 1+ proteinuria on dipstick urinalysis will undergo 24-hour urine collection for quantitative assessment; NOTE: patients with \> 1 g/24 hours will be ineligible
* Clinically significant gastrointestinal malabsorption syndrome
* New York Heart Association congestive heart failure of grade II or above, unstable angina, myocardial infarction within the past 6 months, or serious cardiac arrhythmia associated with significant cardiovascular impairment within the past 6 months; ejection fraction (EF) by multi-gated acquisition (MUGA) or echo should not be less than the institutional lower limit of normal
* Corrected QT (QTc) prolongation \> 480 msec, as calculated by either the Bazett or Fridericia formula, as per institutional standard
* Active hemoptysis (bright red blood \> 1 teaspoon on more than one occasion) =\< 3 weeks prior to registration
* Cohort 2 only: more than one prior treatment with VEGFR active multikinase inhibitor prior to original start of lenvatinib