Erlotinib Hydrochloride and Onalespib Lactate in Treating Patients With Recurrent or Metastatic EGFR-Mutant Non-small Cell Lung Cancer

Inclusion Criteria: * PHASE I: Patients must have metastatic/recurrent, histologically confirmed NSCLC that harbors an EGFR activating mutation (exon 21 L858R, exon 19 deletion, exon 18 G719X, exon 21 L861Q) with progressive disease by RECIST 1.1 on a previous EGFR-tyrosine kinase inhibitor (TKI); OR patients must have metastatic/recurrent histologically confirmed NSCLC that harbors an EGFR exon 20 insertion with progressive disease on platinum containing chemotherapy * PHASE II COHORT A: Patients must have metastatic/recurrent histologically confirmed NSCLC that harbors an EGFR activating mutation (exon 21 L858R, exon 19 deletion, exon 18 G719X, exon 21 L861Q) with stable disease by RECIST 1.1 as best response on erlotinib compared to pre-treatment erlotinib imaging by RECIST 1.1 or progressive disease compared to pre-treatment imaging by RECIST 1.1 after a minimum duration of treatment on erlotinib of 12-weeks; patients must be enrolled within 6 months of initiation of erlotinib * PHASE II COHORT B: Patients must have metastatic/recurrent histologically confirmed NSCLC that harbors an EGFR exon 20 insertion with progressive disease on or after platinum doublet chemotherapy * FOR PHASE I: If patient is on erlotinib at the time of signed consent, the patient does NOT need to be discontinued prior to initiation of erlotinib and onalespib; other EGFR-TKIs must be discontinued at least 7 days prior to initiation of erlotinib and onalespib * FOR PHASE II COHORT A: If patient is on erlotinib at the time of signed consent, erlotinib does NOT need to be discontinued prior to receiving treatment erlotinib and onalespib; last dose of erlotinib must be less than 28 days from when patient signs consent * FOR PHASE II COHORT B: (EGFR exon 20 insertions): Prior EGFR-TKIs including erlotinib is allowed; if patient is on erlotinib at the time of signed consent, erlotinib does NOT need to be discontinued prior to initiation of erlotinib and onalespib * Local testing for EGFR-mutations for this study is acceptable provided it was performed in a Clinical Laboratory Improvement Act (CLIA) certified lab * Patients with a prior history of brain metastases are eligible provided: * The brain metastases have been treated * The patient is asymptomatic from the brain metastases * Corticosteroids prescribed for the management of brain metastases have been discontinued at least 7 days prior to registration * 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 * Measurable disease by RECIST 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin time (PTT) \< 1.3 upper limit of normal (ULN) * 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 and for the duration of study participation; 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 4 months after completion of erlotinib and/or onalespib administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to erlotinib and/or onalespib * History of pneumonitis attributed to an EGFR inhibitor; history of radiation pneumonitis is allowed provided steroid administration for pneumonitis was not required * Mean resting corrected QT interval (QTc using Fridericia's formula \[QTcF\]) \> 470 msec * Left ventricular ejection fraction =\< 50% as demonstrated by echocardiogram or multigated acquisition scan (MUGA) * Drugs that are known to increase torsades de pointes should be avoided; patients must discontinue these medications prior to enrollment on study; selection of alternate concomitant medications with no or minimal torsades de pointes potential is recommended * Strong CYP3A4 inducers and inhibitors should be avoided; selection of alternate concomitant medications with no or minimal CYP3A4 enzyme inhibition potential is recommended; 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 * 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 are excluded from this study; breastfeeding should be discontinued if the mother is treated with erlotinib and onalespib * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible * Prior treatment with a Hsp90 inhibitor * Treatment with proton pump inhibitors within 3 days prior to study entry; if treatment with an histamine (H2)-receptor antagonist such as ranitidine is required, erlotinib must be taken 10 hours after the H2-receptor antagonist dosing and at least 2 hours before the next dose of the H2-receptor antagonist; although the effect of antacids on erlotinib pharmacokinetics has not been evaluated, the antacid dose and the erlotinib dose should be separated by several hours, if an antacid is necessary * Abnormalities of the cornea based on history (e.g., dry eye syndrome, Sjogren's syndrome), congenital abnormality (e.g., Fuch's dystrophy), abnormal slit-lamp examination using a vital dye (e.g., fluorescein, Bengal Rose), and/or an abnormal corneal sensitivity test (Schirmer test or similar tear production test)