Osimertinib and Necitumumab in Treating Patients With EGFR-Mutant Stage IV or Recurrent Non-small Cell Lung Cancer Who Have Progressed on a Previous EGFR Tyrosine Kinase Inhibitor

Inclusion Criteria: * Patients with stage IV or recurrent/metastatic histologically confirmed non-small cell lung cancer (NSCLC) * NSCLC must harbor at least one of the following EGFR activating mutations: Exon 21 L858R, Exon 19 deletion, Exon 18 G719X, Exon 21 L861Q or for EGFR Exon 20 insertion expansion cohort D, NSCLC must harbor an EGFR Exon 20 insertion performed by a Clinical Laboratory Improvement Act (CLIA) certified test * For Dose escalation cohort - progressive disease on at least one prior EGFR-TKI (previous treatment with 3rd generation EGFR-TKI including osimertinib \[AZD9291\] allowed for dose escalation) * For Dose Expansion Cohort A: patient must 1) have progression of disease on erlotinib, gefitinib or afatinib as last previous systemic treatment, 2) have biopsy of tumor taken after progression on erlotinib, gefitinib or afatinib which must be EGFR-T790M negative confirmed by central testing prior to treatment (if EGFR-T790M status is unknown, patients may consent for trial and for biopsy and testing for EGFR T790M will be performed as part of initial biopsy for trial), and 3) be treatment naive to 3rd generation EGFR-TKI (rociletinib, EGFR inhibitor HM61713 \[HM61713\] and osimertinib \[AZD9291\]) and EGFR monoclonal antibodies * For Dose Expansion Cohort B (closed to accrual as of 8/30/18): patient must 1) have progression of disease on a 3rd generation EGFR-TKI such as AZD9291, rociletinib, HM61713, 2) be treatment naive to an EGFR monoclonal antibody, and 3) have a biopsy of tumor taken after progression on last EGFR-TKI that indicates loss of EGFR-T790M (EGFR-T790M negative) confirmed by central testing prior to treatment (if EGFR-T790M status is unknown, patients may consent for trial and for biopsy, and testing for EGFR T790M will be performed as part of initial biopsy for trial) * For Dose Expansion Cohort C: patient must 1) have progression of disease on a 3rd generation EGFR-TKI such as osimertinib (AZD9291), rociletinib, HM61713, 2) be treatment naive to an EGFR monoclonal antibody, 3) have a biopsy of tumor taken after progression on last EGFR-TKI that indicates preservation of EGFR-T790M post-progression on 3rd generation EGFR-TKI with biopsy confirmation by central testing prior to treatment (if EGFR-T790M status is unknown, patients may consent for trial and for biopsy, and testing for EGFR T790M will be performed as part of initial biopsy for trial) * For Dose Expansion Cohort D: patient must 1) tumor that harbors an EGFR Exon 20 insertion by a CLIA certified test, and 2) have progressive disease on or after platinum based chemotherapy, and 3) be treatment naïve to 3rd generation and beyond EGFR-TKI (i.e., osimertinib \[AZD9291\], poziotinib, TAK-778) and EGFR monoclonal antibody; patient who received 1st or 2nd generation EGFR-TKI (such as erlotinib, gefitinib, afatinib) are eligible provided that they did not achieve a response to treatment or they did not have a duration of treatment on EGFR-TKI of 6 months or more * For Dose Expansion Cohort E: patient must have progressive disease on osimertinib (AZD9291) as first-line EGFR-TKI treatment for metastatic NSCLC; patients must also be treatment naive to EGFR-monoclonal antibody * Adequate archival tissue from a biopsy performed after progression of disease on previous EGFR-TKI or willing to consent for a fresh tumor biopsy; (mandatory for Cohorts A, B, C; optional for dose escalation and Cohort D, and Cohort E) * Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1, defined as at least one lesion that can be accurately measured in at least one dimension \>= 10 mm (\>= 1 cm) by computed tomography (CT) imaging or magnetic resonance imaging (MRI) within 42 days prior to registration; the CT from a combined positron emission tomography (PET)/CT may be used only if it is of diagnostic quality; laboratory parameters are not acceptable as the only evidence of disease * Any number of prior therapies is allowed * Age \>= 18 years. NSCLC is exceedingly rare in patients \< 18 years of age. Because no dosing or adverse event data are currently available on the use of and necitumumab in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 * Patients must have the ability to swallow tablets * Life expectancy of greater 3 months * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (patients with Gilbert's syndrome may have serum bilirubin \> 1.5 ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional upper limit of normal * Creatinine =\< 1.5 x ULN OR * Creatinine clearance \>= 50 mL/min * The effects of osimertinib (AZD9291) and necitumumab on the developing human fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception using one of the methods listed below prior to study entry, for the duration of study participation, and for 3 months for women and 6 months for men following the date of the last dose of osimertinib (AZD9291) and/or necitumumab: * Total abstinence from sexual intercourse (minimum one complete menstrual cycle prior to study drug administration); * Vasectomized male subject or vasectomized partner of female subjects * Hormonal contraceptives (oral, parenteral, transdermal or vaginal ring) prior to study drug administration; if the subject is currently using a hormonal contraceptive, she should also use a barrier method during this study and for 3 months after study completion; * Intrauterine device (IUD); * Double-barrier method: male condom plus diaphragm or vaginal cap with spermicide (contraceptive sponge, jellies or creams) * Additionally, for all methods above (except for abstinence), male subjects (including those who are vasectomized) whose partners are pregnant or might be pregnant must use condoms for the duration of the study and for 6 months following completion of therapy * Women of childbearing potential must have a negative urine pregnancy test within 7 days prior to initiation of treatment; women will be considered not of childbearing potential if they are surgically sterile (bilateral oophorectomy or hysterectomy) and/or post menopausal (amenorrheic for at least 12 months); 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 * Patients with untreated brain metastases are allowed provided that the patient is clinically asymptomatic and stable; patients with a prior history of symptomatic brain metastases are eligible provided: * The brain metastases have been treated * The patient is asymptomatic from the brain metastases at enrollment * Corticosteroids prescribed for the management of brain metastases have been discontinued at least 7 days prior to registration * The brain metastases are stable on pre-registration imaging * Patients must have completed last chemotherapy \>= 3 weeks or radiotherapy \>= 2 weeks prior to receiving study drugs * Patients must have recovered from adverse events attributable to previous treatment to =\< grade 1, except for alopecia and sensory neuropathy =\< grade 2 * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Major surgery within 21 days of starting protocol treatment * Patients must discontinue previous EGFR-TKI at least 7 days prior to study enrollment with the exception that patients on osimertinib (AZD9291) for cohorts B, C and E can continue osimertinib (AZD9291) and need not discontinue prior to enrollment * Patients who are receiving any other investigational agents; patients must have discontinued any other investigational agents for at least 5 half-lives or 3 months, whichever is greater, prior to initiation of osimertinib (AZD9291) in an investigational setting * Past medical history of interstitial lung disease, drug-induced interstitial lung disease, radiation pneumonitis requiring steroid treatment, or any evidence of clinically active interstitial lung disease * Patients currently receiving (or unable to stop use prior to receiving the first dose of study treatment) medications or herbal supplements known to be potent inducers of CYP3A4 (at least 3 weeks prior); all patients must try to avoid concomitant use of any medications, herbal supplements and/or ingestion of foods with known inducer effects of CYP3A4 * Patients with active malignancies other than NSCLC or prior curatively treated malignancy at high risk of relapse during the study period with the exception of localized squamous or basal cell skin cancers * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, gastrointestinal disease limiting absorption of osimertinib (AZD9291) such as a malabsorption syndrome or inflammatory bowel disease or psychiatric illness/social situations that would limit compliance with study requirements * Mean resting corrected QT interval (QTc using Fridericia's formula \[QTcF\]) \> 470 msec * Any clinically important abnormalities in rhythm, conduction or morphology of resting electrocardiography (ECG) (e.g., complete left bundle branch block, third degree heart block, second degree heart block) * Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives or any concomitant medication known to prolong the QT interval and cause torsades de pointes * Left ventricular ejection fraction \< 50% on echocardiogram or multi-gated acquisition (MUGA) * The effects of osimertinib (AZD9291) and necitumumab on the developing human fetus are unknown; for this reason and because EGFR inhibitors are known to be teratogenic, pregnant women are excluded from this study; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother osimertinib (AZD9291) and necitumumab breastfeeding should be discontinued if the mother is treated with osimertinib (AZD9291) and necitumumab; these potential risks may also apply to other agents used in this study * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with osimertinib (AZD9291)