MK2206 and Erlotinib Hydrochloride in Treating Patients With Advanced Non-Small Cell Lung Cancer Who Have Progressed After Previous Response to Erlotinib Hydrochloride Therapy

Inclusion Criteria:

* Patients must have histologically or cytologically confirmed non-small cell lung cancer of any histologic subtype

  * NOTE: epidermal growth factor receptor (EGFR) mutational status (either wild-type or positive for an activating mutation) will be determined for all patients on this study; commercial assays for EGFR mutation status are allowed; knowledge of EGFR mutational status is not required at the time of protocol entry but should be determined or known before the end of course 2; however, if one of the strata is temporarily closed to accrual, knowledge of EGFR mutational status will be required prior to protocol entry
* Patients may have measurable or non-measurable disease; x-rays and/or scans for disease assessment of measurable disease must have been completed within 28 days prior to registration
* Patients must have radiologic or clinical progressive disease following prior benefit (response or stable disease) to EGFR-tyrosine kinase inhibitor (TKI) therapy (e.g., erlotinib) administered either as a single agent or in combination with other agents for at least 12 weeks prior to progression; Note: patients may have received intervening systemic therapy after EGFR-TKI progression); additionally, patients must have documentation of radiographic progression within the preceding three months prior to study entry
* Prior cytotoxic chemotherapy is allowed; any number of prior chemotherapy regimens is also allowed; prior cetuximab therapy is also allowed; NOTE: a patient with an EGFR activating mutation who has received EGFR-TKI therapy as first line therapy, but has not received platinum-based chemotherapy, would be considered eligible for this trial
* Karnofsky performance status \>= 60%
* Absolute neutrophil count (ANC) \>= 1,500/mcL
* Platelet count \>= 100,000/mcL
* Total bilirubin =\< upper institutional normal 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 =\< upper institutional normal limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal
* Prior to the first patient registration, this study must be institutional review board approved; a copy of the institutional review board (IRB) approval for each site involved must be given to the Data Coordinating Center at City of Hope
* Women of childbearing potential and men must use two forms of 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, the patient should inform the treating physician immediately
* Patients on coumadin should have their international normalized ratio (INR) monitored at least once per week or more frequently depending on the investigator's judgment; there have been some case reports of increased INR when coumadin is co-administered with erlotinib
* Ability to understand and the willingness to sign a written informed consent document
* Patients should have tumor tissue (either fresh frozen tumor tissue or paraffin-embedded tumor tissue) available for retrieval; if an endobronchial lesion is present or suspected, bronchoscopy is recommended as a source of fresh tissue; tissue blocks or unstained slides from the time of original diagnosis are acceptable if repeat biopsy is not feasible

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 any ongoing grade 2 or greater toxicity from a prior treatment
* Patients may not be receiving any other investigational agents
* Patients with symptomatic brain metastases should be excluded from this clinical trial; patients with asymptomatic controlled or treated (e.g., with radiation and/or surgery) brain metastases are otherwise eligible as long as corticosteroids given expressly for brain metastases (mets) have been stopped for at least 14 days
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-2206 or erlotinib
* Caution must be observed for patients receiving any medications or substances that are strong inhibitors or inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP 450 3A4); although these patients are still potentially eligible, close monitoring is required for toxicity
* Preclinical studies demonstrated the potential of MK-2206 for induction of hyperglycemia in all preclinical species tested; patients with diabetes or in risk for hyperglycemia should not be excluded from trials with MK-2206, but the hyperglycemia should be well controlled on oral agents before the patient enters the trial
* Preclinical studies indicated transient changes in corrected QT (QTc) interval during MK-2206 treatment; prolongation of QTc interval is potentially a safety concern while on MK-2206 therapy; cardiovascular: baseline Fridericia QT (QTcF) \> 450 msec (male) or QTcF \> 470 msec (female) will exclude patients from entry on study
* 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 this combination
* Human immunodeficiency (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Prior MK-2206 therapy is not allowed
* Patients unable to swallow MK-2206 tablets and erlotinib tables whole are ineligible; (the tablets cannot be crushed or broken)