TLR8 Agonist VTX-2337 and Cetuximab in Treating Patients With Locally Advanced, Recurrent, or Metastatic Squamous Cell Cancer of Head and Neck

Inclusion Criteria: * Patients with a histological or cytopathological confirmed diagnosis of squamous cell carcinoma of the head and neck region that is: * Locally advanced/recurrent and no longer amenable to local surgical or radiation therapy and/or * Has evidence of metastatic disease * Patients may have been previously treated with systemic therapy but are otherwise deemed currently platinum-refractory, or would be deemed inappropriate or intolerant to platinum-based chemotherapy * Patients must have completed definitive chemotherapy and/or radiation therapy \>= 3 months prior to study entry * Prior therapy with agents targeting/blocking the epidermal growth factor receptor (e.g. cetuximab and erlotinib) is allowable * Performance Status: Eastern Cooperative Oncology Group (ECOG) 0 - 2 * Expected life expectancy of at least 12 weeks, as assessed by the Investigator * Ability and willingness to comply with the study's visit and assessment schedule and to provide voluntary written informed consent * Absolute neutrophil count (ANC) \>= 1,500 cells/μL * Platelet count \>= 75,000 cells/μL * Hemoglobin \>= 8.0 g/dL * Creatinine =\< 2.0 mg/dL * Total bilirubin =\< 2.0 x upper limit of normal (ULN) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]), serum glutamic pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x ULN * For patients with liver metastases, AST, ALT \< 5x ULN is acceptable * Willingness to use a medically acceptable method of contraception throughout the study period and for 4 weeks after the final administration of VTX-2337 (all subjects) * For female subjects with reproductive potential: a negative serum pregnancy test Exclusion Criteria: * Investigational therapy within 4 weeks of study entry * Chemotherapy therapy or palliative radiation therapy within the previous 2 weeks prior to dosing with cetuximab or VTX-2337; patients should have recovered from major toxicities of prior therapy (If deemed reversible, toxicities should return to baseline or =\< grade 2 in severity) * Major surgery within the past 4 weeks prior to dosing with cetuximab or VTX-2337 * Concurrent symptomatic central nervous system (CNS) involvement, brain or leptomeningeal metastases; treated CNS involvement which has been stable \> 28 days off systemic steroids may be included * Major active psychiatric disorders which would limit compliance * Treatment with oral or parenteral corticosteroids within 2 weeks prior to dosing with VTX-2337 or a requirement for systemic immunosuppressive therapy for any reason * Active autoimmune disease * Clinically significant cardiac disease (e.g., congestive heart failure, unstable or uncontrolled angina, myocardial infarction) within 6 months of dosing with VTX-2337 * Clinically significant ophthalmologic disease, defined as: * Current retinal vascular disorder, including active untreated diabetic retinopathy and/or * Previous or current uveitis * Infection requiring parenteral antibiotic therapy or causing fever (temp \> 100.5 degrees Fahrenheit \[F\] or 38.1 degrees Celsius \[C\]) within 1 week prior to dosing with VTX-2337 * Pregnant or breast-feeding females * Uncontrolled inter-current illness, pre-planned surgery or procedure requiring hospitalization during the study period, or any other condition or circumstance that could interfere with adherence to the study's procedures or requirements, or otherwise compromise the study's objectives * Second primary malignancy that is clinically detectable (not including in situ carcinoma of the cervix, non-melanoma skin cancer or low-grade \[Gleason score =\< 6\] localized prostate cancer) and demonstrating active progression at the time of consideration for study enrollment * Known prior severe allergic/hypersensitivity to cetuximab or any of the components of the study treatment * Known prior severe (\>= Grade 3) rash and / or diarrhea toxicities to cetuximab