Dabrafenib With or Without Trametinib in Treating Patients With Advanced Differentiated Thyroid Cancer

Inclusion Criteria:

* Patients must have histologically or cytologically confirmed papillary thyroid cancer, follicular thyroid cancer (tall cell variant, insular thyroid cancer, follicular variant of papillary thyroid cancers, poorly differentiated thyroid cancer or any of the above mixed histology will be allowed)
* Presence of BRAF mutation in tumor tissue
* 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 computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam; malignant lymph nodes will be considered measurable if they are \>= 15 mm in short axis
* Patients must have progressive disease within the thirteen months prior to study entry; progressive disease is as defined in Response Evaluation Criteria in Solid Tumors (RECIST) 1.1, which is at least a 20% increase in the sum of diameters of target lesions and the sum must also demonstrate an absolute increase of at least 5 mm; the appearance of one or more new lesions is also considered progressive disease
* Patients are willing to have tumor biopsy pre-study and at 4 weeks on study (fine needle aspiration or core biopsy) if patient has biopsy-accessible tumors as determined by investigator
* Patients must have disease that is refractory (unresponsive) to radioactive iodine (RAI) treatment as defined by one of the following:

  * One or more measurable lesions that do not demonstrate RAI uptake
  * One or more measurable lesions progressive by RECIST 1.1 within 12-months of prior RAI therapy
  * Cumulative RAI dose of \> 600 mCi
* Prior therapy allowed:

  * Patients may have been previously treated with up to three regimens of oral multikinase inhibitors, including sorafenib, sunitinib and pazopanib
  * Patients may have been previously treated with external beam radiation or cytotoxic chemotherapy therapy
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Absolute neutrophil count \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Total bilirubin =\< 1.5 X institutional upper limit of normal
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal
* Serum creatinine =\< 1.5 X institutional upper limit of normal
* Left ventricular ejection fraction (EF) \>= institutional lower limit of normal
* Patient must have a calcium phosphate product (CPP) =\< 4.0 mmol\^2/L\^2 (50 mg\^2/dL\^2)
* Female patients of childbearing age are required to have a negative serum pregnancy test within 14 days prior to the first dose of study medication

  * Females are required to use an effective method of contraception from the time of negative serum pregnancy test, throughout the study duration, and until 4 weeks after the last dose of study medication; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her 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 for 16 weeks after completion of the last dose of study drug
  * Specific contraception requirements for females: female subjects of childbearing potential must not become pregnant and are required to be sexually inactive by abstinence or use contraceptive methods with a failure rate of \< 1%; sexual inactivity by abstinence must be consistent with the preferred and usual lifestyle of the subject; periodic abstinence (e.g. calendar, ovulation, symptothermal, post ovulation methods) and withdrawal are not acceptable methods of contraception; contraceptive methods with a failure rate of \< 1% include the following:

    * Intrauterine device (IUD) or intrauterine system (IUS) that meets the \< 1% failure rate as stated in the product label
    * Male partner sterilization (vasectomy with documentation of azoospermia) prior to the female subject's entry into the study, and this male is patient's sole sexual partner; for this definition, "documented" refers to the outcome of the investigator's/qualified physician designee's medical examination of the subject or review of the subject's medical history for study eligibility, as obtained via a verbal interview with the subject or from the subject's medical records
    * Double barrier method: condom and occlusive cap (diaphragm or cervical/vault caps) plus spermicidal agent (foam/gel/film/cream/suppository); these allowed methods of contraception are only effective when used consistently, correctly and in accordance with the product label; the investigator is responsible for ensuring subjects understand how to properly use these methods of contraception
  * Specific contraception requirements for males: to prevent pregnancy in a female partner or to prevent exposure of any partner to the investigational product from a male subject's semen, male subjects must use one of the following contraceptive methods during the study and for a total of 16 weeks following the last dose of study drug (based upon the lifecycle of sperm):

    * Abstinence, defined as sexual inactivity consistent with the preferred and usual lifestyle of the subject for 14 days prior to first dose of study drug, through the dosing period, and for at least 16 weeks after the last dose of study drug; periodic abstinence (e.g. calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception
    * Condom (during non-vaginal intercourse with any partner - male or female) OR
    * Condom and occlusive cap (diaphragm or cervical/vault caps) plus spermicidal agent (foam/gel/film/cream/suppository) (during sexual intercourse with a female)
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients had external beam radiotherapy, cytotoxic chemotherapy, or oral multikinase inhibitors within 4 weeks prior to entering the study
* Patients who have been treated with radioactive iodine within 24 weeks prior to entering the study (radioactive iodine within 4 weeks will be allowed if negative post-treatment scan)
* Patients have not recovered from adverse events related to prior chemotherapy, radiation therapy or multikinase inhibitors to Common Terminology Criteria for Adverse Events (CTCAE) 4.0 grade 1 or less except for alopecia
* Patients have been previously treated with potent BRAF inhibitor or MEK inhibitor, including PLX4032/vemurafenib, ARQ 736; previous treatment with sorafenib is permitted
* Patients are receiving any other investigational agents
* Patients are on any medication that is on the list of prohibitive medications; patients on therapeutic dose of warfarin; this is due to potential for significant interactions between warfarin and study agents
* Patients with a known history of infection with hepatitis B virus (HBV) or hepatitis C virus (HCV)
* Patients with a history of other malignancy; patients who have been disease-free from other malignancy for 5 years or greater, or patients with a history of resected non-melanoma skin cancer, or patients with a history of treated in situ carcinoma will be allowed
* Patients with uncontrolled brain metastases; patients who are on a stable dose of corticosteroids for more than 1 month or off corticosteroids for 2 weeks can be enrolled; enzyme-inducing anti-epileptic drugs are not permitted
* Patients with a known history of retinal vein occlusion or central serous retinopathy, or predisposing factors to retinal vein occlusion (RVO) or central serous retinopathy (CSR) (e.g. uncontrolled glaucoma or ocular hypertension, uncontrolled systemic disease such as hypertension, diabetes mellitus, or history of hyperviscosity or hypercoagulability syndromes)
* Visible retinal pathology as assessed by ophthalmic exam that is considered a risk factor for RVO or CSR such as:

  * Evidence of new optic disc cupping
  * Evidence of new visual field defects
  * Intraocular pressure \> 21 mm Hg as measured by tonography
* Patients with a known history of glucose-6-phosphate dehydrogenase (G6PD) deficiency
* Patients with class II, III, or IV heart failure as defined by the New York Heart Association (NYHA) functional classification system

  * Abnormal cardiac valve morphology (subjects with minimal abnormalities, can be entered on study with approval)
* Corrected QT (QTc) interval greater than or equal to 480 msecs (\>= 500 msec for subjects with bundle branch block)
* Patients with 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
* Pregnant women and nursing women are excluded from this study
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Subjects with a history of pneumonitis or interstitial lung disease
* History of acute coronary syndromes (including myocardial infarction and unstable angina), coronary angioplasty, or stenting within the past 6 months
* History of uncontrolled arrhythmias; subjects with controlled atrial fibrillation for \> 1 month prior to study day 1 are eligible