Trametinib in Increasing Tumoral Iodine Incorporation in Patients With Recurrent or Metastatic Thyroid Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants); confirmation of thyroid carcinoma will be done at Memorial Sloan-Kettering (MSK) * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral CT scan, magnetic resonance imaging (MRI), or calipers by clinical exam; tumors in previously irradiated fields may be considered measurable if there is evidence of tumor progression after radiation treatment * RAI-refractory disease on structural imaging, defined as any one of the following: * A metastatic lesion that is not radioiodine-avid on a diagnostic radioiodine scan performed prior to enrollment in the current study, or * A radioiodine-avid metastatic lesion which remained stable in size or progressed despite radioiodine treatment 6 months or more prior to entry in the study; there are no size limitations for the index lesion used to satisfy this entry criterion * The presence of at least one fluorodeoxyglucose (FDG) avid lesion * No recent treatment for thyroid cancer as defined as: * No prior RAI therapy is allowed \< 6 months prior to initiation of therapy on this protocol; a diagnostic study using \< 10 millicurie (mCi) of RAI is not considered RAI therapy * No external beam radiation therapy \< 4 weeks prior to initiation of therapy on this protocol; (previous treatment with radiation for any indication is allowed if the investigator judges that the previous radiation does not significantly compromise patient safety on this protocol) * No chemotherapy or targeted therapy (e.g., tyrosine kinase inhibitor) is allowed \< 4 weeks prior to the initiation of therapy on this protocol * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 3 months * Able to swallow and retain orally-administered medication and does not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels * All prior treatment-related toxicities must be Common Terminology Criteria for Adverse Events version 4.0 (CTCAE v 4.0) grade =\< 1 (except alopecia); grade 2 prior treatment related toxicities may be allowed after discussion with the principal investigator * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Hemoglobin \>= 9 g/dL * Platelets \>= 100 x 10\^9/L * Albumin \>= 2.5 g/dL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x institutional ULN * Creatinine =\< 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault formula) \>= 50 mL/min OR 24-hour urine creatinine clearance \>= 50 mL/min * Prothrombin time (PT) or international normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.5 x institutional ULN * Left ventricular ejection fraction \>= institutional lower limit of normal (LLN) by echocardiogram (ECHO) or multi gated acquisition scan (MUGA) * The effects of trametinib on the developing human fetus are unknown; since MEK inhibitors and RAI may be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study, and 4 months after study completion; women of child-bearing potential must have a negative pregnancy test within 2 weeks prior study registration; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform the treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of trametinib administration * Ability to understand and the willingness to sign a written informed consent document * Patients must agree to undergo two separate biopsies of a malignant lesion; biopsies do not need to be done if one of the following apply: * If either the site investigator or person performing the biopsy judges that no tumor is accessible for biopsy or that biopsy poses too great of a risk to the patient (if the only tumor accessible for biopsy is also the only lesion that can be used for RECIST v1.1 response evaluation, then the patient may be exempt from biopsy after discussion with the MSK principal investigator) * The goal will be to have a minimum of 6 patients from Cohort A and 3 patients from Cohort B attempt to have one or both of these research biopsies done (for a total of 9 patients total); accrual may be limited only to subjects for whom tumor is accessible for biopsy and attempt at biopsy is considered safe if continued enrollment of those who are not candidates for biopsy make it impossible to reach the accrual goals for research biopsies described above (e.g., if 19 \[of 25\] patients are accrued to Cohort A without any biopsies having been obtained within the cohort, then all further subjects who are registered to that cohort must qualify for attempted research biopsy in order to be enrolled into the study \[i.e., subjects who would have been excluded from having biopsies done due to the above reasons would be excluded from participating in the study; these conditions also apply to Cohort B\]) * Availability of archival tumor tissue from the thyroid cancer primary or metastasis (a tissue block or a minimum of 30 unstained slides would be required; patients with less archival tissue available may still be eligible for the study after discussion with the MSK principal investigator) * COHORT A: Confirmation in a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory that one of the patient's thyroid tumors (primary tumor, recurrent tumor, or metastasis) has an neuroblastoma RAS viral (v-ras) oncogene homolog (NRAS) or Kirsten rat sarcoma viral oncogene homolog (KRAS) or Harvey rat sarcoma viral oncogene homolog (HRAS) mutation at G12, G13, or Q61; this group of patients will also be referred to as "RAS MUT" * COHORT A: Patients must have progressive disease, defined as the presence of new or growing lesion(s) on radiologic imaging within 14 months of study enrollment and/or new/worsening disease related symptoms within 14 months of study enrollment; (progression according to RECIST v 1.1 criteria is not required) * COHORT B: Confirmation in a CLIA certified laboratory that one of the patient's thyroid tumors (primary tumor, recurrent tumor, or metastasis) does not have any of the following mutations: * Mutation at V600 of the v-raf murine sarcoma viral oncogene homolog B (BRAF) gene * Mutation in NRAS or KRAS or HRAS at G12, G13, or Q61 * These patients will be designated "BRAF/RAS wild type (WT)" Exclusion Criteria: * History of another malignancy; exception: patients who have been disease-free for 3 years, patients with a history of completely resected non-melanoma skin cancer, and/or patients with indolent secondary malignancies, are eligible; MSK can consult the Cancer Therapy Evaluation Program (CTEP) Medical Monitor if unsure whether second malignancies meet the requirements specified above * History of interstitial lung disease or pneumonitis * Use of other investigational drugs within 28 days (or five half-lives, whichever is shorter; with a minimum of 14 days from the last dose) preceding the first dose of trametinib and during the study * Symptomatic or untreated leptomeningeal or brain metastases or spinal cord compression * Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO) or to thyrotropin alpha (Thyrogen) * Current use of a prohibited medication; the following medications or non-drug therapies are prohibited: * Other anti-cancer therapy while on study; (note: megestrol \[Megace\] if used as an appetite stimulant is allowed; thyroid-stimulating hormone \[TSH\] suppressive therapy is also allowed; palliative radiation therapy to non-target lesions is also allowed) * Concurrent treatment with bisphosphonates is permitted; prophylactic use of bisphosphonates in patients without bone disease is not permitted, except for the treatment of osteoporosis * Because the composition, pharmacokinetics (PK), and metabolism of many herbal supplements are unknown, the concurrent use of all herbal supplements is prohibited during the study (including, but not limited to, St. John's wort, kava, ephedra \[ma huang\], gingko biloba, dehydroepiandrosterone \[DHEA\], yohimbe, saw palmetto, or ginseng) * Patients with the following ophthalmological findings/conditions: * Intraocular pressure \> 21 mmHg, or uncontrolled glaucoma (irrespective of intraocular pressure) * Current or past history of central serious retinopathy or retinal vein occlusion * History or evidence of cardiovascular risk including any of the following: * Left ventricular ejection fraction (LVEF) \< LLN * A QT interval corrected for heart rate using the Bazett's formula (QTcB) \>= 480 msec * History or evidence of current clinically significant uncontrolled arrhythmias (exception: patients with controlled atrial fibrillation for \> 30 days prior to randomization are eligible) * History of acute coronary syndromes (including myocardial infarction and unstable angina), coronary angioplasty, or stenting within 6 months prior to randomization * History or evidence of current \>= class II congestive heart failure as defined by the New York Heart Association (NYHA) functional classification system * Treatment-refractory hypertension defined as a blood pressure of systolic \> 140 mmHg and/or diastolic \> 90 mmHg which cannot be controlled by anti-hypertensive therapy * Patients with intra-cardiac defibrillators * Known cardiac metastases * Known human immunodeficiency virus (HIV), hepatitis B virus (HBV), or hepatitis C virus (HCV) infection (with the exception of chronic or cleared HBV and HCV infection, which will be allowed); patients with human immunodeficiency virus (HIV) are not eligible if on anti-retroviral medications * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Animal reproductive studies have not been conducted with trametinib; therefore, the study drug must not be administered to pregnant women or nursing mothers; these potential risks may also apply to RAI and Thyrogen * HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with trametinib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy * Patients unable to follow a low iodine diet or requiring medication with high content in iodide (amiodarone) * Patients who received iodinated intravenous contrast as part of a radiographic procedure within 3 months of study registration; those that have had iodinated intravenous contrast within this time frame may still be eligible if a urinary iodine analysis reveals that the excess iodine has been cleared after the last intravenous contrast administration