Erlotinib in Combination With Select Tyrosine Kinase Inhibitors in Adult Patients With Advanced Solid Tumors

* INCLUSION CRITERIA: * Patients must have histologically confirmed solid tumors that have progressed on standard therapy known to prolong survival or for which no standard treatment options exist. * Age \>=18 years. * Patients must have evaluable disease according to RECIST 1.1 criteria. * ECOG performance status =\< 2. * Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count \>=1,500/mcL * Platelets \>=100,000/mcL * Total bilirubin \<=1.5 X institutional ULN (with the exception of those with Gilbert syndrome, who must have total bilirubin \<=3 X institutional ULN * AST(SGOT)/ALT(SGPT) \<=3 X institutional upper limit of normal; \<= 5.0 x ULN in patients with liver metastases * creatinine \<=1.5 X institutional ULN OR * creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels \>1.5 mg/dL * Based on preclinical safety data and their respective mechanisms of action, erlotinib, lenvatinib, and axitinib can cause fetal harm when administered to pregnant women. For this reason, women of child-bearing potential and men enrolled on this protocol must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 1 month after dosing with study drugs ceases. 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. * Potential trial participants should have recovered from clinically significant adverse events of their most recent therapy/intervention prior to enrollment. * Patients with non-healing wounds/fistulas may enroll if no immediate surgical input is required in the opinion of the PI. * Patients must have \<= 1+ proteinuria on urinalysis, or \< 1 g protein on 24-hour urine collection, or a urine protein:creatinine ratio of \< 1. * Prior anti-EGFR- and anti-VEGF/VEGFR-targeted therapy is permitted, provided that the patient has not undergone prior anti-EGFR/anti-VEGF(R) TKI combination therapy. * Patients must be able to swallow. * Patients with treated brain metastases are eligible if follow-up brain imaging after CNS-directed therapy shows no evidence of progression. * Patients with new or progressive 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 known to be positive for HIV who meet the following criteria will be considered eligible: * CD4 count \> 350 cells/mm\^3 * Undetectable viral load for 6 months prior to enrollment * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents * No history of AIDS-defining opportunistic infections * Patients must be willing to provide blood for research purposes. EXCLUSION CRITERIA: * For the erlotinib-lenvatinib arm only: patients with a QTcF interval of \>=480 msec at study entry or with congenital long QT syndrome are excluded. * Patients who are receiving any investigational agents are excluded. * Patients who are receiving \>2 anti-hypertensive agents will be excluded. * Patients who are receiving strong CYP3A4- and/or CYP1A2-inhibiting or -inducing agents that cannot be discontinued or replaced with an alternative medication will be excluded. * Patients who are receiving agents that increase gastric pH and that cannot be discontinued or replaced with an alternative medication will be excluded. * Patients who smoke tobacco will be excluded. * Patients with a history of cirrhosis will be excluded if found to have a moderate or severe Child-Pugh score. * Patients should not, in the opinion of the Principal Investigator, have GI impairment that may limit the absorption of erlotinib, lenvatinib, or axitinib. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study drugs. * Uncontrolled intercurrent illness that would, in the opinion of the Principal Investigator, limit compliance with study requirements. * Pregnant and breastfeeding women are excluded from this study because all 3 study drugs can cause fetal harm, based on preclinical safety data and the respective drug mechanisms of action. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with the study drugs, breastfeeding should be discontinued prior to the first dose of study drug, and women should refrain from nursing throughout the treatment period and for 1 month following the last dose of study drug.