Erlotinib Hydrochloride and Cetuximab in Treating Patients With Advanced Gastrointestinal Cancer, Head and Neck Cancer, Non-Small Cell Lung Cancer, or Colorectal Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed incurable gastrointestinal tract, head and neck, or non-small cell lung cancers that are KRAS wild type; if KRAS mutational status cannot be determined on archived tumor tissue from the patient, a needle or excisional biopsy of a malignant site may be performed prior to enrollment; mutational status may be determined either by polymerase chain reaction (PCR) assay (e.g., DxS KRAS mutation kit) or by direct sequencing of KRAS exon 2, codons 12 and 13; the result must detect no mutations at these sites * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * 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 transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 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 * Eligibility of patients receiving any medications or substances known to affect or with the potential to affect the activity or pharmacokinetics of erlotinib will be determined following review of their case by the principal investigator; although concomitant use of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers is not prohibited in this study, identification of MTD and DLT may be affected by their use; concomitant use of any of these drugs will be noted in the case report forms and will be taken into account in determining MTD and DLT of this therapy; efforts should be made to switch patients with a history of brain metastases who are taking enzyme-inducing anticonvulsant agents to other medications * 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 participating in this study, she should inform her treating physician immediately * Other prior malignancies are allowed provided prior therapy has been discontinued and there is no evidence of disease (NED) * Patients must be able to take and retain oral medications * 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 may not be receiving any other investigational agents * Patients with a history of brain metastases are eligible provided that the metastases have been surgically resected and/or are radiographically and clinically stable for 2 months following the completion of radiation therapy * History of allergic reactions attributed to compounds of similar chemical or biologic composition to erlotinib * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cetuximab * Prior treatment with EGFR-targeting therapies * Major surgery or significant traumatic injury occurring within 21 days prior to treatment * Abnormalities of the cornea based on history (e.g., dry eye syndrome, Sjogren's syndrome), congenital abnormality (e.g., Fuch's dystrophy), abnormal slit-lamp examination using a vital dye (e.g., fluorescein, Bengal-Rose), and/or an abnormal corneal sensitivity test (Schirmer test or similar tear production test) * Gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for IV alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease * 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 erlotinib or cetuximab * Human immunodeficiency virus (HIV)-positive patients receiving combination anti-retroviral therapy are excluded from the study