Testing of the Anti Cancer Drugs CB-839 HCl (Telaglenastat) and MLN0128 (Sapanisertib) in Advanced Stage Non-small Cell Lung Cancer

Inclusion Criteria: * Patients must have stage IV or recurrent/metastatic NSCLC and have progressed on or after platinum-based chemotherapy and/or PD-(L)1 immune checkpoint inhibitor. Patients with autoimmune or other conditions where PD-(L)1 checkpoint inhibitors are contraindicated are eligible with progression on or after platinum-based chemotherapy or immunotherapy * Dose escalation: patients with NSCLC known to harbor EGFR, ALK, ROS1, BRAF V600E/K activating mutations must have also progressed on appropriate Food and Drug Administration (FDA)-approved targeted therapies to be eligible for dose escalation * Dose expansion: patients must have stage IV or recurrent/metastatic NSCLC harboring 1) NFE2L2 mutations (LSCC); 2) KEAP1 mutations (LSCC); KRAS/KEAP1 or KRAS/NFE2L2 co-mutations (non-squamous NSCLC); or 3) LSCC WT for NFE2L2 or KEAP1 who have progressed on or after platinum-based chemotherapy and/or PD (L)1 immune checkpoint inhibitors or immunotherapy. Acceptable molecular testing includes Foundation ACT circulating tumor deoxyribonucleic acid (ctDNA) or Guardant 360 ctDNA in plasma or Foundation One or Memorial Sloan Kettering Cancer Center (MSK)-Integrated Mutation Profiling of Actionable Cancer Targets (IMPACT) in tumor tissue. Patients with autoimmune or other conditions where PD-(L)1 checkpoint inhibitors are contraindicated are eligible with progression on or after platinum-based chemotherapy or immunotherapy * Eastern Cooperative Oncology Group (ECOG) - American College of Radiology Imaging Network (ACRIN) performance status 0-2 * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Fasting blood glucose (FBS) =\< 130 and hemoglobin A1C (HGBA1C) =\< 8.0% and fasting triglycerides =\< 300 mg/dL * Tissue accessible for fine needle and/or core needle biopsy for molecular testing (for expansion cohorts only) * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN (=\< 5 x ULN if liver metastases are present) * Creatinine =\< 1.3 mg/dL OR * Glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 * Hemoglobin \>= 9 g/dL (without transfusion within 1 week preceding study drug administration) * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Because no dosing or adverse event data are currently available on the use of CB-839 HCl (telaglenastat) in combination with MLN0128 (sapanisertib) in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with stable, treated, asymptomatic brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Females of childbearing potential must have a negative pregnancy test (=\< 14 days) prior to start of trial treatment * Females 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 effective method of contraception and 1 additional effective (barrier) method, at the same time, from the time of signing the informed consent through 90 days (or longer as mandated by local labeling \[e.g. USPI, SmPC, etc\]) 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 patient (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) * Agree not to donate egg(s) during the course of this study or within 90 days after receiving their last dose of study drug * Male patients, even if surgically sterilized (i.e. status post vasectomy), must agree to the following contraceptive requirements: * Agree to practice highly 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 patient (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) * Agree not to donate sperm during the course of this study or within 120 days after receiving their last dose of study drug * The effects of CB-839 HCl (telaglenastat) and MLN0128 (sapanisertib) on the developing human fetus are unknown. For this reason, 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 CB-839 HCl (telaglenastat) administration * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Patients who have had chemotherapy within 3 weeks or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to CB-839 HCl (telaglenastat) and MLN0128 (sapanisertib) * CB-839 HCl (telaglenastat) is a weak in vitro inhibitor of CYP2C9. Therefore, patients receiving any medications or substances that are substrates of CYP2C9 are eligible, but should use caution with substrates that have a narrow therapeutic index. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. 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 * Patients with uncontrolled intercurrent illness * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because CB-839 HCl (telaglenastat) is an agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events (AEs) in nursing infants secondary to treatment of the mother with CB-839 HCl (telaglenastat), breastfeeding should be discontinued if the mother is treated with CB-839 HCl (telaglenastat). These potential risks may also apply to MLN0128 (sapanisertib) * Patients who are unable to swallow tablets * Human immunodeficiency virus (HIV)-infected patients * Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of MLN0128 (sapanisertib). In addition, patients with enteric stomata are also excluded * 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 * Medically significant (symptomatic) bradycardia * History of arrhythmia requiring an implantable cardiac defibrillator * Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated demonstration of QTc interval \> 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes) * Patients receiving systemic corticosteroids (either IV or oral steroids, excluding inhalers or low-dose hormone replacement therapy) within 1 week before administration of the first dose of study drugs * Patients who are taking proton pump inhibitors (PPIs) within 7 days of the first dose of study drug or who require treatment with PPIs throughout the trial or those who are taking H2 receptor antagonists within 24 hours of the first dose of study drug * Previous treatment with PI3K, AKT, dual PI3K/mTOR inhibitors, TORC1/2 inhibitors or TORC1 inhibitors * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better