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

• 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