Combination Chemotherapy With or Without Erlotinib Hydrochloride in Treating Patients With Metastatic or Recurrent Squamous Cell Carcinoma of the Head and Neck

Inclusion Criteria: * Histologically confirmed metastatic or recurrent squamous cell carcinoma of the head and neck (SCCHN) of the oral cavity, oropharynx, hypopharynx or larynx; metastatic or recurrent lesions of the nasopharynx and sinus are excluded * Radiologically measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan; measurable lymph nodes are required to be \>= 15 mm in size (short axis diameter) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) =\< 2 * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelet count \>= 100 x 10\^9/L * Total bilirubin =\< upper limit of normal (ULN) (excluding Gilbert's disease) * Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 1.5 x ULN * Alkaline phosphatase =\< 2.5 x ULN * Serum creatinine =\< 1.5 x ULN * Patients with reproductive potential (e.g., females menopausal for less than 1 year and not surgically sterilized) must practice effective contraceptive measures for the duration of study drug therapy and for at least 30 days after completion of study drug therapy; female patients of childbearing potential must provide a negative pregnancy test (serum or urine) =\< 14 days prior to treatment initiation * Written informed consent to participate in the study according to the investigational review board (IRB) or independent ethics committee (IEC) Exclusion Criteria: * Histology other than squamous cell carcinoma * Primary sites other than oral cavity, oropharynx, hypopharynx, and larynx * Prior palliative chemotherapy for metastatic or recurrent disease * Prior biological therapy for metastatic or recurrent disease within 3 weeks prior to randomization * Patients with known, untreated brain metastases; patients with treated (irradiated or resected) brain metastases are eligible if treatment was completed more than 28 days prior to study entry and if clinical neurologic function is stable * Pre-existing peripheral neuropathy \>= grade 2 * History of poorly controlled gastrointestinal disorders that could affect the absorption of the study drug (e.g., Crohn's disease, ulcerative colitis); patients requiring feeding tubes are permitted * Other active malignancies requiring chemotherapy treatment within 2 years prior to randomization, except for adequately treated basal cell or squamous cell skin cancer or in situ cervical or breast cancer or superficial, resected melanoma * Serious underlying medical condition which would impair the ability of the patient to receive protocol treatment, in the opinion of the treating physician * History of allergic reactions to compounds of similar chemical composition to the study drugs (docetaxel, cisplatin, carboplatin, erlotinib or their excipients), or other drugs formulated with polysorbate 80 * Any concurrent anti-cancer therapy, excluding hormonal therapy for prostate or breast cancer * Dementia or significantly altered mental status that would prohibit the understanding and giving of informed consent * Women who are pregnant or breast-feeding and women or men not practicing effective birth control