Afatinib After Chemoradiation and Surgery in Treating Patients With Stage III-IV Squamous Cell Carcinoma of the Head and Neck at High-Risk of Recurrence

Inclusion Criteria: * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Patients must have pathological evidence of persistent primary and/or lymph node disease with viable tumor cells following primary concurrent chemoradiotherapy of locoregionally advanced (stage III/IV) HNSCC of the oral cavity, oropharynx, larynx, hypopharynx or p16-positive unknown primary of the head and neck region; persistent primary and/or lymph node disease with viable tumor cells will be determined by the histological determination of tumor viability defined as tumor cells with intact cellular compartments (i.e. cytoplasm and nucleus) that do not exhibit karyolysis, pyknosis, or karyorrhexis on haematoxylin and eosin (H\&E) staining * Patients must have undergone a neck dissection following completion of chemoradiotherapy and must have involved at the minimum a compartment dissection of nodal levels with residual abnormalities on post-treatment imaging studies and/or received a complete resection of the residual primary lesion with negative margins * Patients must have achieved a complete response at the primary disease site after chemoradiotherapy or complete resection of the residual primary site disease with negative margins * All persistent lymph node disease must have received at least 66 Gy of radiotherapy and must have been completely surgically resected prior to registration, and surgical incisions should be adequately healed * Patients with extracapsular lymph node extension, perineural or lymphovascular invasion will be eligible * Patients must be at least 6 weeks (42 days) and no more than 36 weeks (252 days) from completion of chemoradiation at the time of registration * Patients will be eligible regardless of ability to swallow; patients with dysphagia may have afatinib/placebo administered via gastrostomy tube * Absolute neutrophil count \>= 1,000/mm\^3 * Platelets \>= 100,000/mm\^3 * Total bilirubin =\< 1.5 x the upper limit of normal (ULN) * Aspartate amino transferase (AST) =\< 3 x the ULN * Alanine amino transferase (ALT) =\< 3 x the ULN * Calculated creatinine clearance must be \>= 50 ml/min using the Cockcroft-Gault formula * Prior cetuximab or any epidermal growth factor receptor (EGFR) inhibitors will not be permitted including cetuximab administered with a chemoradiotherapy or radiotherapy regimen * As all patients in this study will have received prior full dose, curative-intent external-beam radiotherapy to the involved neck, no additional external-beam radiotherapy will be permitted prior to or during study participation * Women of childbearing potential and sexually active males must be strongly advised to use an accepted and effective method of contraception * Patients must have electrocardiogram (ECG) within 8 weeks prior to randomization to the study * Patients must be assessed for cardiac function by echocardiogram (ECHO) or multi-gated acquisition scan (MUGA) within 8 weeks prior to randomization Exclusion Criteria: * Patients with known distant metastatic disease or with any gross residual disease following salvage primary tumor resection or neck dissection * Known hypersensitivity to afatinib or any of the excipients of this product * Prior adjuvant chemotherapy (aside from the initial induction chemotherapy followed by chemoradiotherapy or chemoradiotherapy regimen) * History of acute myocardial infarction within 3 months prior to registration, and any history of uncontrolled angina, uncontrolled arrhythmia, or uncontrolled heart failure * Pregnant or breast-feeding women * Active infections, other cancers, or history of other cancers * Participation in any other clinical trials or taking any other experimental medications * Left ventricular dysfunction * Evidence of interstitial lung disease