Azacitidine and Entinostat Before Chemotherapy in Treating Patients With Advanced Non-small Cell Lung Cancer

Inclusion Criteria: * Patients must have histologically or cytologically proven non-small cell lung cancer; tumor tissue must be available from all patients prior to initiation of protocol therapy, either from original diagnostic biopsy, or biopsy performed prior to initiation of protocol therapy * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have received 1 prior platinum containing doublet regardless of mutation status * Patients with targetable mutation i.e. epidermal growth factor receptor (EGFR) or anaplastic lymphoma kinase (ALK), must have been treated with at least 1 prior tyrosine kinase inhibitor (TKI) * Prior immunotherapy is allowed * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky \>= 70%) * Life expectancy of greater than 12 weeks * 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 transferase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transferase \[SGPT\]) =\< 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 * 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 * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients who are receiving any other investigational agents * Patients with uncontrolled brain metastases; patients with brain metastases must have stable neurologic status following local therapy (surgery or radiation) for at least 4 weeks, and must be without neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; patients may be treated with steroids as clinically indicated * Patients with liver metastases that replace greater than 30% of the liver parenchyma * History of allergic reactions attributed to compounds of similar chemical or biologic composition to entinostat, azacitidine, mannitol, irinotecan, docetaxel, pemetrexed, or gemcitabine, or other agents used in the study * Uncontrolled inter-current illness including, but not limited to, ongoing or active infection, New York Heart Association (NYHA) class 3-4 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 on this protocol * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible