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, subf…