Osimertinib in Combination With Alisertib or Sapanisertib for the Treatment of Osimertinib-Resistant EGFR Mutant Stage IIIB or IV Non-Small Cell Lung Cancer

Inclusion Criteria:

* Patients must have histologically or cytologically confirmed non-small cell lung cancer
* Stage IIIB/IV or recurrent non-small cell lung cancer which is not amenable to curative intent therapy
* EGFR exon 21 L858R or exon 19 deletion mutation, or T790M mutation that was acquired following treatment with first or second generation tyrosine kinase inhibitor (TKI). Eligible EGFR mutation must be confirmed by Clinical Laboratory Improvement Amendments (CLIA) certified test
* Patients must have either a) disease progression on osimertinib within 30 days prior to study enrollment. Patients who continued osimertinib beyond disease progression (e.g. patients with oligo-progression who had radiation) may be eligible on further disease progression after discussion with principal investigator OR b) disease progression on osimertinib and one other line of systemic therapy (if the other systemic therapy line included PD-L1 blockade then the last dose of the latter must be more than 3 months prior to study enrollment). The number of patients who had prior osimertinib and other line of systemic therapy will be capped at 25% in each study arm
* Local ablative therapy (e.g. stereotactic radiosurgery \[SRS\], stereotactic body radiation therapy \[SBRT\], or surgery) for brain or systemic metastases prior to enrollment is allowed
* Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Absolute neutrophil count (ANC) \> 1500/ mm\^3 (within 28 days before the first dose of study drug)
* Platelets \> 100,000/mm\^3 (within 28 days before the first dose of study drug)
* Hemoglobin \> 9 g/dL. Values must be obtained without need for myeloid growth factor or transfusion support within 14 days, however, erythrocyte growth factor is allowed as per published American Society of Clinical Oncology (ASCO) guidelines (within 28 days before the first dose of study drug)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (within 28 days before the first dose of study drug)
* Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase \[ALT\]) \< 2.5 x ULN (may be up to 5 x ULN if with known liver metastases \[mets\]) (within 28 days before the first dose of study drug)
* ALISERTIB ARM: Creatinine clearance (Cockcroft-Gault Formula) \>= 30 ml/min (within 28 days before the first dose of study drug)
* SAPANISERTIB ARM: Creatinine clearance (Cockcroft-Gault Formula) \>= 60 ml/min (within 28 days before the first dose of study drug)
* SAPANISERTIB ARM: Glycosylated hemoglobin (HbA1c) \< 7.0% (within 28 days before the first dose of study drug)
* SAPANISERTIB ARM: Fasting serum glucose (FSG) =\< 130 mg/dL (within 28 days before the first dose of study drug)
* SAPANISERTIB ARM: Fasting triglycerides (TG) =\< 300 mg/dL (within 28 days before the first dose of study drug)
* Willing to provide blood and tissue for correlative research purposes
* Female patients who:

  * Are postmenopausal for at least 1 year before the screening visit, OR
  * Are surgically sterile, OR
  * If they are of childbearing potential, agree to practice 1 highly effective method of contraception and 1 additional effective (barrier) method at the same time, from the time of signing the informed consent through 180 days after the last dose of study drug, OR

    * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\], withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.)
* Male patients, even if surgically sterilized (i.e., status postvasectomy), who:

  * Agree to practice effective barrier contraception during the entire study treatment period and through 120 days after the last dose of study drug, OR
  * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\], withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.)
* Voluntary written consent must be given before performance of any study related procedure not part of standard of care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care
* Ability to swallow oral medications
* Resolution of all acute toxic effects of prior treatments (chemotherapy, immunotherapy, radiotherapy, surgical procedures) to grade 1 or less (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version 5.0). Patients should have tolerated osimertinib 80 mg PO daily with no current grade 2 or greater adverse events (AE) attributable to osimertinib

Exclusion Criteria:

* ALISERTIB ARM: Radiation therapy to more than 25% of the bone marrow. Whole pelvic radiation is considered to be over 25%
* ALISERTIB ARM: Prior allogeneic bone marrow or organ transplantation
* Known gastrointestinal (GI) disease or GI procedures that could interfere with the oral absorption or tolerance of study drugs. Examples include, but are not limited to partial gastrectomy, history of small intestine surgery, and celiac disease. In addition, patients with enteric stomata are also excluded
* Inability to swallow oral medication or inability or unwillingness to comply with the administration requirements related to study drugs
* ALISERTIB ARM: Known history of uncontrolled sleep apnea syndrome and other conditions that could result in excessive daytime sleepiness, such as severe chronic obstructive pulmonary disease; requirement for supplemental oxygen
* Requirement for constant administration of proton pump inhibitor, H2 antagonist, or pancreatic enzymes throughout the study. The intermittent use of proton pump inhibitor (PPI), H2-antagonists and antacids (including carafate) is only allowed within the following guidelines:

  * H2 receptor antagonists until 24 hours (h) of the first dose of study drug
  * Antacid formulations until 2 hours before dosing and after 2 hours following dosing.
  * PPI is allowed until 5 days before the first study treatment dose. PPIs are prohibited throughout the study
* SAPANISERTIB ARM: Patients receiving systemic corticosteroids (either intravenous \[IV\] or oral steroids, excluding inhalers or low-dose hormone replacement therapy) within 1 week before administration of the first dose of study drug
* History of any of the following within the last 6 months before administration of the first dose of the drug:

  * New York Heart Association (NYHA) class III or IV heart failure, ischemic myocardial event, including angina requiring therapy and artery revascularization procedures
  * Ischemic cerebrovascular event, including transient ischemic attack and artery revascularization procedures
  * Pulmonary embolism
* Any of the following cardiac criteria

  * Mean resting corrected QT interval (QTc using Fridericia's formula) \> 470 msec or torsades de pointes
  * Any clinically important abnormalities in rhythm, conduction or morphology of resting electrocardiogram (ECG) e.g., complete left bundle branch block, third degree heart block, second degree heart block, PR interval \> 250 msec. Prior to study entry, any ECG abnormality at Screening has to be documented by the investigator as not medically relevant.
  * 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
* SAPANISERTIB ARM: Significant active cardiovascular or pulmonary disease including:

  * Uncontrolled hypertension (i.e., systolic blood pressure \> 180 mm Hg, diastolic blood pressure \> 95 mm Hg). Use of anti-hypertensive agents to control hypertension before cycle 1 day 1 is allowed
  * Pulmonary hypertension
  * Uncontrolled asthma or oxygen (O2) saturation \< 90% by arterial blood gas analysis or pulse oximetry on room air
  * Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement
* Past medical history of interstitial lung disease (ILD), drug-induced ILD, radiation pneumonitis which required steroid treatment, or any evidence of clinically active interstitial lung disease
* Patients with uncharacterized eye disorders
* Unstable central nervous system (CNS) metastasis (as defined by need for steroids in last 14 days)
* Patients who are currently receiving treatment with contraindicated QTc prolonging medications (list provided https://crediblemeds.org/pdftemp/pdf/CombinedList.pdf) or potent CYP3A4 inducers/inhibitors (list provided https://www.fda.gov/drugs/drug-interactions-labeling/drug-development-and-drug-interactions-table-substrates-inhibitors-and-inducers), if that treatment cannot be either discontinued or switched to a different medication prior to first day of study treatment
* Known leptomeningeal carcinomatosis
* SAPANISERTIB ARM: Known hepatitis B surface antigen-positive, or known or suspected active hepatitis C infection
* ALISERTIB ARM: Known human immunodeficiency virus (HIV) positive patients who meet the following criteria will be considered eligible:

  * CD4 count \> 350 cells/mm\^3
  * Undetectable viral load
  * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents
  * No history of acquired immunodeficiency syndrome (AIDS)-defining opportunistic infections
* SAPANISERTIB ARM: Known HIV positive patients
* Use of investigational drugs within 30 days or 5 half-lives of enrollment (whichever was greater)
* ALISERTIB ARM: Previous treatment with aurora kinase inhibitors
* SAPANISERTIB ARM: Previous treatment with PI3K, AKT, dual PI3K/mTOR inhibitors, TORC1/2 inhibitors or TORC1 inhibitors
* Diagnosed or treated for another malignancy within 2 years of enrollment, with the exception of complete resection of basal cell carcinoma or squamous cell carcinoma of the skin, an in situ malignancy, or low-risk prostate cancer after curative therapy
* Other severe acute or chronic medical or psychiatric condition, or laboratory abnormality that may increase the risk associated with study participation or investigational product administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the patient inappropriate for enrollment in this study
* Female subject who is pregnant or breast-feeding.

  * Confirmation that the subject is not pregnant must be established by a negative serum beta-human chorionic gonadotropin (beta-hCG) pregnancy test result obtained during screening. Pregnancy testing is not required for post-menopausal or surgically sterilized women.
* Female patient who intend to donate eggs (ova) during the course of this study or 180 days after receiving their last dose of study drug(s)
* Male patients who intend to donate sperm during the course of this study or 4 months after receiving their last dose of study drug(s)