Auranofin and Sirolimus in Treating Patients With Advanced Solid Tumors or Recurrent Non-Small Cell Lung Cancer

Inclusion Criteria: * Cohort I (dose escalation) only: Histologic proof of an advanced, solid tumor that is now unresectable * Cohort II (maximum tolerated dose) only: * Platinum-refractory non-small cell lung cancer (NSCLC) (platinum-refractory defined as either disease progression either during or within 6 months of completion of first-line platinum-based chemotherapy) * Measurable disease * Evidence of disease progression =\< 6 months prior to registration * Received at least one prior approved chemotherapeutic regimen unless there is no known, approved therapeutic regimen for their malignancy * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * Platelets (PLT) \>= 100,000/mm\^3 * Total bilirubin =\< 1.5 upper limit of normal (ULN) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 3 x ULN (=\< 5 x ULN for patients with liver involvement) * Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 3 x ULN (=\< 5 x ULN for patients with liver involvement) * Creatinine =\< 1.5 x ULN * Fasting blood glucose =\< 126 mg/dL * Fasting triglycerides =\< 1.5 x ULN * Fasting cholesterol =\< 1.5 x ULN * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Ability to provide informed written consent * Willing to return to Mayo Clinic in Florida for follow-up (during the active monitoring phase of the study); note: during the active monitoring phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up * Life expectancy \>= 84 days (3 months) * Willing to provide tissue and blood samples for correlative research purposes; note: the goals of this study include assessment of the biologic effects on surrogate markers of the agent(s) being tested and are, therefore, contingent upon availability of the biologic specimens * Negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only Exclusion Criteria: * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy * 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 * Any of the following prior therapies: * Chemotherapy =\< 28 days prior to registration * Mitomycin C/nitrosoureas =\< 42 days prior to registration * Immunotherapy =\< 28 days prior to registration * Biologic therapy =\< 28 days prior to registration * Radiation therapy =\< 28 days prior to registration * Radiation to \> 25% of bone marrow * Bevacizumab =\< 28 days prior to registration * Failure to fully recover from acute, reversible effects of prior chemotherapy regardless of interval since last treatment * New York Heart Association classification III or IV cardiovascular disease * Central nervous system (CNS) metastases or seizure disorder; NOTE: patients with known brain metastases that have been successfully treated and stable for \>= 6 months without requirement for corticosteroids and without seizure activity will be eligible * Receiving therapeutic anticoagulation with warfarin; NOTE: prophylactic anticoagulation (i.e., low dose warfarin) of venous or arterial access devices is allowed, provided that international normalized ratio (INR) \< 1.5; therapeutic anti-coagulation with low molecular weight heparin is allowed at time of registration * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Immunocompromised patients and/or patients known to be human immunodeficiency virus (HIV) positive * Cohort II Only: Other active malignancy =\< 5 years prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history or prior malignancy, patient must not be receiving other cytotoxic or molecularly targeted therapeutics treatment for their cancer; patients receiving certain hormonal manipulations as part of their treatment may be allowed to continue at the discretion of the principal investigator (PI) (e.g. luteinizing hormone-releasing hormone \[LHRH\] analogs for prostate cancer); concurrent endocrine therapy for breast cancer will not be permitted * History of myocardial infarction or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias =\< 168 days (6 months) prior to registration * Known allergy to auranofin or other gold compounds * Receiving any medications or substances that are strong inhibitors or strong inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4); NOTE: use of strong inhibitors and inducers is prohibited =\< 7 days prior to registration