Cetuximab With or Without Tivantinib in Treating Patients With Head and Neck Cancer That Is Recurrent, Metastatic, or Cannot Be Removed by Surgery

Inclusion Criteria: * Histologically/cytologically confirmed diagnosis of squamous cell carcinoma of head and neck origin not amenable to curative intent therapy; both human papillomavirus (HPV) positive (+) and HPV negative (-) are eligible, but status has to be known prior to randomization (although not required for consenting); any type of tissue based HPV assessment is acceptable (e.g. p16 immunohistochemistry \[IHC\] or HPV in situ hybridization \[ISH\]); if local HPV testing is not available slides can be sent to the University of Chicago for HPV testing; please note that p16 IHC is generally only considered to be accurate for oropharyngeal tumors * Presence of measurable lesions (as per Response Evaluation Criteria in Solid Tumors \[RECIST\] 1.1); generally a \>= 10 mm tumor lesion (in the longest diameter by computed tomography \[CT\] scan) or a lymph node \>= 15 mm (short axis) is considered measurable disease when evaluated by CT scan (with a slice thickness no greater than 5 mm) * Availability of tissue (10 tumor containing formalin-fixed, paraffin-embedded \[FFPE\] slides/sections) * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 * Patients who have received cetuximab or another inhibitor of epidermal growth factor receptor (EGFR) in the curative intent treatment setting (e.g. with radiation or during induction chemotherapy \[prior to definitive, curative intent therapy\]) are eligible for the study * Life expectancy of greater than 8 weeks * Hemoglobin \>= 9.0 g/dL * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Serum creatinine =\< 1.5 x institutional upper limit of normal OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Patients must be able to swallow ARQ 197 (tivantinib) by mouth, unless adequate data about administration by gastrostomy (G)-tube becomes available; tablets may be crushed, but must be taken orally * Human immunodeficiency virus (HIV)-positive patients with normal immune function (cluster of differentiation \[CD\]4 count \> 200) are eligible if there are no drug interactions with ARQ 197 (tivantinib) or cetuximab * 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 she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of ARQ 197 (tivantinib) administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 2 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 2 weeks earlier * Nasopharyngeal tumors that show lymphoepithelioma histology * Patients who have received more than 2 prior cytotoxic treatments in the palliative treatment setting are ineligible * Patients who have received treatment with an EGFR or MET inhibitor in the palliative treatment setting are ineligible * Patients with known, active brain metastases should be excluded from this clinical trial; patients with treated brain metastases stable for \>= 12 weeks are eligible; use of corticosteroid (for patients with brain metastasis and other indications for corticosteroid use) is acceptable on a low maintenance or tapering dose schedule * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ARQ 197 (tivantinib) or cetuximab * Concurrent life-threatening diseases: patients with diseases which with reasonable certainty do not limit life expectancy to 12 months or less are eligible; assessment of such concurrent illnesses should be by the principal investigator * 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 ARQ 197 (tivantinib) * Concurrent use of warfarin (therapeutic use) is allowed, but requires close monitoring of prothrombin time (PT)/international normalized ratio (INR) * History of congestive heart failure defined as class II to IV per New York Heart Association (NYHA) classification; active coronary artery disease (CAD), clinically significant bradycardia or other uncontrolled, cardiac arrhythmia defined as \>= grade 3 according to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, or uncontrolled hypertension; myocardial infarction occurring within 6 months prior to study entry (myocardial infarction occurring \> 6 months prior to study entry is permitted) * Patients may not be receiving any other investigational agents