Dabrafenib and Lapatinib in Treating Patients With Refractory Thyroid Cancer That Cannot Be Removed by Surgery

Inclusion Criteria: * Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative measures do not exist or are no longer effective * Patients must have measurable and histologically or cytologically confirmed thyroid cancer with a BRAF V600E or V600K (c. 1799 T to A and c.1799\_1800TG\>AA) mutation that is not considered curable by surgery; confirmation will be done at Memorial Sloan Kettering (MSK); only tumors with a BRAFV600E or BRAFV600K mutation will be eligible for the clinical study; BRAF status will be assessed in a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory; BRAF status may also be tested with any Food and Drug Administration (FDA)-approved test (such as Cobas 4800 BRAF V600 Mutation Test) * The tumor is considered to be radioactive-iodine refractory by any of the following criteria: * Total lifetime dose of radioactive iodine \> 600 mCi * Absent or insufficient radioactive iodine uptake in either all lesions or an index lesion which has never been resected or received external beam radiation therapy as documented on a radioactive iodine scan (insufficient uptake must be confirmed by either an endocrinologist or nuclear medicine physician) * Progression of disease (by imaging or thyroglobulin) within 6 months of radioactive iodine treatment * Fludeoxyglucose F 18 (FDG)-avid lesion (standard uptake variable maximum \[SUVmax\] \>= 3) on a FDG-positron emission tomography (PET) scan * No recent treatment for thyroid cancer as defined as: * No radioactive iodine therapy is allowed if given \< 3 months prior to initiation of this protocol therapy; a diagnostic study using \< 10 mCi of radioactive iodine (RAI) is not considered radioactive iodine therapy * No external beam radiation therapy \< 4 weeks prior to initiation of therapy on this protocol * No chemotherapy or targeted therapy (e.g., tyrosine kinase inhibitor) is allowed \< 4 weeks prior to the initiation of therapy * Age \>= 18 years * Because no dosing or adverse event data are currently available on the use of dabrafenib in combination with lapatinib in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 or Karnofsky \>= 60% * Life expectancy of greater than 2 months * Able to swallow and retain oral medication and must not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels * Absolute neutrophil count (ANC) \>= 1.2 x 10\^9/L, within 2 weeks of the first dose of study treatment * Hemoglobin \>= 9 g/dL, within 2 weeks of the first dose of study treatment * Platelets \>= 100 x 10\^9/L, within 2 weeks of the first dose of study treatment * Bilirubin =\< 1.5 x institutional upper limit of normal (ULN) except subjects with known Gilbert's syndrome, within 2 weeks of the first dose of study treatment * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x institutional ULN, within 2 weeks of the first dose of study treatment * Blood creatinine =\< 1.5 mg/dL (if blood creatinine is \> 1.5 mg/dL, calculate creatinine clearance using standard Cockcroft and Gault method or using a 24 hour urine collection for creatinine; creatinine clearance must be \> 50 mL/min), within 2 weeks of the first dose of study treatment * Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.3 x institutional ULN; subjects receiving anticoagulation treatment may be allowed to participate with PT/INR/PTT established within the therapeutic range prior to randomization; subjects will be eligible if it is determined by a hematologist that the cause is not associated with clinical bleeding (e.g., deficiency of factor XII), within 2 weeks of the first dose of study treatment * Left ventricular ejection fraction \>= institutional lower limit of normal (LLN) by echocardiogram (ECHO), within 2 weeks of the first dose of study treatment * Women of childbearing potential must have a negative serum pregnancy test within 7 days of the first dose of study treatment * Pregnancy and breast feeding * The effects of dabrafenib on the developing human fetus are unknown; for this reason and because other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (barrier method of birth control, or abstinence; hormonal contraception is not allowed due to drug-drug interactions which can render hormonal contraceptives ineffective) for the duration of study participation, and for at least 2 weeks after treatment with dabrafenib; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately * Based on studies in animals, it is also known that dabrafenib may cause damage to the tissue that makes sperm. This may cause sperm to be abnormal in shape and size and could lead to infertility, which may be irreversible * Dabrafenib should not be administered to pediatric populations outside clinical trials * Ability to understand and the willingness to sign a written informed consent document * Patient must agree to allow 3 separate biopsies of any malignant lesion; biopsies do not need to be done if: * Tumor is not considered accessible by either the investigator or the person performing the biopsy (it is determined the risk is too high due to location near vital organs or too great of a risk of an adverse event) * Patient is on anticoagulation and it would be unsafe to temporarily hold the anticoagulation * Consent of the principal investigator (PI) not to have a biopsy done * A minimum of 8 subjects must participate in the biopsy part of the study Exclusion Criteria: * Prior systemic anti-cancer therapy (chemotherapy with delayed toxicity, extensive radiation therapy, immunotherapy, biologic therapy, or vaccine therapy) within the last 3 weeks; chemotherapy regimens without delayed toxicity within the last 2 weeks preceding the first dose of study treatment * 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 study treatment and during the study * Current use of a prohibited medication; patients receiving any medications or substances that are strong inhibitors or inducers of cytochrome P450, family 3, subfamily A (CYP3A) or cytochrome P450, family 2, subfamily C, polypeptide 8 (CYP2C8) are ineligible; current use of, or intended ongoing treatment with: herbal remedies (e.g., St. John's wort), or strong inhibitors or inducers of P-glycoprotein (Pgp) or breast cancer resistance protein 1 (Bcrp1) should also be excluded; because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated list of these agents; medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Prohibited: strong inducers of CYP3A or CYP2C8, since concentrations of dabrafenib may be decreased * Antibiotics: rifamycin class agents (e.g., rifampin, rifabutin, rifapentine) * Anticonvulsant: carbamazepine, oxcarbazepine phenobarbital, phenytoin, s-mephenytoin * Miscellaneous: bosentan, St. John's wort * Prohibited: strong inhibitors of CYP3A or CYP2C8, since concentrations of dabrafenib may be increased * Antibiotics: clarithromycin, telithromycin, troleandomycin * Antidepressant: nefazodone * Antifungals: itraconazole, ketoconazole, posaconazole, voriconazole * Hyperlipidemia: gemfibrozil * Antiretroviral: ritonavir, saquinavir, atazanavir * Miscellaneous: conivaptan * Unresolved toxicity of National Cancer Institute Common Terminology Criteria for Adverse Events, version 4.0 (NCI CTCAE v4.0) grade 2 or higher from previous anti-cancer therapy, except alopecia; in specific cases, will be allowed with permission from the principal investigator * Human immunodeficiency virus (HIV)-positive patients on antiviral drugs and/or cluster of differentiation (CD)4 count is inadequate (\< 500); if neither condition exists, HIV-positive patients are eligible * 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 * A history of hepatitis B virus (HBV) or hepatitis C virus (HCV) infection (with the exception of cleared HBV and HCV infection, which will be allowed) * Presence of an invasive malignancy other than the study indication under this trial within 3 years of study enrollment except for carcinoma in situ CIS, squamous cell carcinomas of the skin, or basal cell carcinoma of the skin; a diagnosis of an invasive malignancy within 3 years is allowed if both the cure rate is felt to be \> 80% and there has been no evidence of disease in the past year * Patients with a history of RAS mutation-positive tumors are not eligible regardless of interval from the current study; Note: prospective RAS testing is not required; however if the results of previous RAS testing are known, they must be used in assessing eligibility * Brain metastases that are symptomatic or requiring corticosteroids (except inhaled); subjects must also be off of enzyme-inducing anticonvulsants for \> 4 weeks * History or evidence of cardiovascular risks including any of the following: * History of acute coronary syndromes (including myocardial infarction or unstable angina), coronary angioplasty, or stenting within the past 24 weeks prior to randomization * History or evidence of current class II, III, or IV heart failure as defined by the New York Heart Association (NYHA) functional classification system * Intra-cardiac defibrillators * Abnormal cardiac valve morphology (\>= grade 2) documented by ECHO; (subjects with grade 1 abnormalities \[i.e., mild regurgitation/stenosis\] can be entered on study); subjects with moderate valvular thickening should not be entered on study * History or evidence of current clinically significant uncontrolled cardiac arrhythmias; clarification: subjects with atrial fibrillation controlled for \> 30 days prior to dosing are eligible * Treatment refractory hypertension defined as a blood pressure of systolic \> 140 mmHg and/or diastolic \> 90 mm Hg which cannot be controlled by anti-hypertensive therapy * Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to the study treatments, their excipients, and/or dimethyl sulfoxide (DMSO) * Medical or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because of the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with dabrafenib, breastfeeding should be discontinued if the mother is treated with dabrafenib; these potential risks may also apply to other agents used in this study