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

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

What this study is about