Cetuximab and Everolimus in Treating Patients With Metastatic or Recurrent Colon Cancer or Head and Neck Cancer

Inclusion Criteria:

* Patients must have histologically-confirmed advanced solid tumors
* Patients who are refractory to standard therapy; patients with metastatic, irinotecan-refractory colon cancer, or recurrent/metastatic head and neck cancer may enroll, as cetuximab monotherapy is among the standard options for such patients; patients with locally advanced, treatment-naïve head and neck cancer who are candidates for radiation with cetuximab are not eligible, as radiation provides them a survival benefit, and the number of projected cetuximab doses would be only seven
* Development of new lesions or an increase in preexisting lesions on bone scintigraphy, computed tomography (CT), magnetic resonance imaging (MRI) or by physical examination; patients in whom the sole criterion for progression is an increase in a biochemical marker, e.g., carcinoembryonic antigen (CEA), or an increase in symptoms, are not eligible
* No radiotherapy (unless palliative), treatment with cytotoxic agents, or treatment with biologic agents =\< 3 weeks prior to registration on this study (6 weeks for mitomycin or nitrosoureas); \>= 2 weeks must have elapsed from any prior surgery or hormonal therapy; patients must have fully recovered from the acute toxicities of any prior treatment with cytotoxic drugs, radiotherapy or other anti-cancer modalities (returned to baseline status as noted before most recent treatment); patients with persisting, stable chronic toxicities from prior treatment =\< grade 1 are eligible
* Eastern Cooperative Oncology Group (ECOG) performance status =\<2 (Karnofsky \>= 60%)
* Life expectancy of \> 3 months
* Hemoglobin \>= 9 g/dL
* Leukocytes \>=3 K/mm\^3
* Absolute neutrophil count \>= 1.5 K/mm\^3
* Platelets \>= 100 K/mm\^3
* Total bilirubin within institutional normal limits
* Hepatitis B panel negative
* Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamic pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal (ULN)
* Creatinine within 1.5 x ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal
* Women of child-bearing potential (WOCBP) must agree to use adequate contraception (hormonal or barrier method of birth control; complete abstinence) prior to study entry and for the duration of study participation and for 3 months after the conclusion of study therapy, and must have a negative serum or urine pregnancy test =\< 7 days prior to registration; pregnant and nursing patients are excluded because the side effects of the combination of cetuximab and RAD001 on a fetus or nursing child are unknown; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; sexually active men must also use appropriate contraception method and should not father a child while receiving therapy during this study
* Ability to understand and the willingness to sign a written informed consent document
* Fasting serum cholesterol \< 350 mg/d L and triglycerides \< 400 mg/d L
* Able and willing to undergo pharmacokinetic (PK) and pharmacodynamic (PD) testing as outlined in this protocol; if however the tumor is not amenable to the PD requirements of the protocol, the patient must be willing and able to undergo skin biopsy

Exclusion Criteria:

* Chronic treatment with systemic steroids or another immunosuppressive agent
* Uncontrolled brain or leptomeningeal metastases, including patients who continue to require glucocorticoids for brain or leptomeningeal metastases
* Impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of RAD001 (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection)
* Patients with an active, bleeding diathesis or on therapeutic anticoagulation (except low dose coumadin)
* Patients may not have received prior cetuximab therapy
* Patients may not be receiving any other investigational agents; in addition, patients must not have received investigational treatment =\< 30 days prior to registration
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to agents used in study
* Patients with chronic active hepatitis B or recent hepatitis B infection (hepatitis B surface antigen \[HepB sAg\] or immunoglobulin M \[IgM\] antibody to hepatitis B core antigen \[IgM antiBc\] positive) are ineligible because these patients are at increased risk of reactivation of the hepatitis B virus which may be fatal due to the immunosuppressive properties of RAD001
* Known human immunodeficiency virus (HIV)-positive patients are ineligible because these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy and the potential pharmacokinetic interaction between antiretroviral therapy and the investigational agents
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, oxygen dependent pulmonary disease or psychiatric illness/social situations that would limit compliance with study requirements; other concurrent severe and/or uncontrolled medical disease which could compromise participation in the study (i.e., uncontrolled diabetes, uncontrolled hypertension, severe malnutrition, active ischemic heart disease, myocardial infarction within six months, chronic liver or renal disease, active upper gastrointestinal \[GI\] tract ulceration)
* All WOCBP MUST have a negative pregnancy test =\< 7 days prior to registration; if the pregnancy test is positive, the patient must not receive investigational product and must not be enrolled in the study