Pembrolizumab and Afatinib in Patients With Non-small Cell Lung Cancer With Resistance to Erlotinib

Inclusion Criteria: * Incurable, advanced or metastatic/recurrent non-small cell lung cancer with EGFR activating mutations (exon 19 del, exon 21 L858R, L861Q, G718X); who have radiologic and/or clinically progressive disease on erlotinib at any point during the patient's cancer treatment as determined by the Investigator * Be willing and able to provide written informed consent for the trial * Have a life expectancy of at least 3 months * Have measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Adequate archival tissue for determination of EGFR-mutation status and PD-L1 status with a leftover cell block (or equivalent) for additional immune correlates from a tumor lesion biopsied in the last 60 days that has not been previously irradiated occurring: 1) after progression on erlotinib and no intervening systemic treatment between biopsy and initiation of MK-3475 and afatinib or amenable to repeat biopsy * Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) (Zubrod) performance scale * Be willing to consent for biopsy at baseline (if inadequate archival tissue per inclusion criteria above) and an on treatment biopsy; have a tumor in a location that in the opinion of the investigator that is amenable to biopsy or have provided tissue for PD-L1 and other biomarker analysis from a newly obtained (within 60 days) formalin fixed tumor tissue from a recent biopsy of a tumor lesion not previously irradiated; no systemic antineoplastic therapy may be administered between the PD- L1 biopsy and initiating study medication; fine needle aspirates are not acceptable; needle or excisional biopsies, or resected tissue is required; the tissue sample must be received by the central vendor (Labcorp) and evaluated for adequacy prior to starting therapy * There is no limit to the number of prior treatments for this phase I trial * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L * Serum creatinine OR measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) =\< 1.5 x upper limit of normal (ULN) OR \>= 60 mL/min for subject with creatinine levels \> 1.5 x institutional ULN * Creatinine clearance should be calculated per institutional standard * Serum total bilirubin =\< 1.5 x ULN * 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 * International normalized ratio (INR) or prothrombin time (PT) =\< 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; 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 * Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Female subjects of childbearing potential should be willing 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; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Male subjects should agree 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 Exclusion Criteria: * Is currently participating in or has participated in a study of an investigational agent or using an investigational device within 4 weeks of the first dose of treatment * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment * Has had a prior monoclonal antibody within 4 weeks prior to study day 1 or who has not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier; denosumab is allowed as long as not \< 1 week prior to study day 1 and not administered on day of MK-3475 infusion * Has had prior chemotherapy or targeted small molecule therapy within 2 weeks prior to study day 1 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 * Note: if subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy * Has a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy * Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis; 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 at least 7 days prior to trial treatment * Has an active autoimmune disease requiring systemic treatment within the past 3 months or a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids or immunosuppressive agents; subjects with vitiligo or resolved childhood asthma/atopy would be an exception to this rule; subjects that require intermittent use of bronchodilators or local steroid injections would not be excluded from the study; subjects with hypothyroidism stable on hormone replacement or Sjorgen's syndrome will not be excluded from the study * Has evidence of interstitial lung disease or active, non-infectious pneumonitis that required oral or intravenous glucocorticoids to assist with management; lymphangitic spread of the non-small cell lung cancer (NSCLC) is not exclusionary * Has an active infection requiring intravenous systemic therapy * Has a 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 * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * History or presence of clinically relevant cardiovascular abnormalities such as uncontrolled hypertension, congestive heart failure New York Heart Association (NYHA) classification of 3, unstable angina or poorly controlled arrhythmia as determined by the investigator; myocardial infarction within 6 months prior to enrollment * Is 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 * Has received prior therapy with an anti-programmed cell death (PD)-1, anti-PD-L1, anti-PD-L2, anti-cluster of differentiation (CD)137, or anti-cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (including ipilimumab or any other antibody or drug specifically targeting T-cell co-stimulation or checkpoint pathways) * Known hypersensitivity to afatinib or the excipients of any of the trial drugs * Prior participation in an afatinib clinical study, even if not assigned to afatinib treatment * Has a known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies) * Has 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) * Has received a live vaccine within 30 days prior to the first dose of trial treatment * The presence of poorly controlled gastrointestinal disorders that could affect the absorption of the afatinib (e.g. Crohn's disease, ulcerative colitis, chronic diarrhea, malabsorption) * Subjects that require treatment with a strong inhibitor of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) will be excluded; they may be included if there is an alternate treatment available (not a strong CYP3A4 inhibitor) and they are willing to switch prior to enrollment; if a subject opts to change from a strong CYP 3A4 inhibitor to a weaker CYP 3A4 inhibitor, the subject must stop the strong CYP 3A4 inhibitor 7 days before study drug administration * Receiving drugs known to be strong inducers or inhibitors of permeability (P)-glycoprotein that are known to interact with afatinib including, but not limited to: ritonavir, cyclosporine A, ketoconazole, itraconazole, erythromycin, verapamil, quinidine, tacrolimus, nelfinavir, saquinavir, and amiodarone; 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; the subject must stop the strong inducer or inhibitor of P-glycoprotein 7 days before or 5 half lives before study drug administration (whichever timepoint is longer) * Major surgery within 4 weeks before starting study treatment or scheduled for surgery during the projected course of the study * Hormonal treatment within 2 weeks prior to start of study treatment * Radiotherapy within 4 weeks prior to enrollment, except as follows: * Palliative radiation to target organs other than chest may be allowed up to 2 weeks prior to enrollment, and * Single dose palliative treatment for symptomatic metastasis outside above allowance to be discussed with sponsor-investigator prior to enrolling * Major surgery within 4 weeks before starting study treatment or scheduled for surgery during the projected course of study