• Patients must have histologically or cytologically confirmed American Joint Committee on Cancer (AJCC) stage III-IV T0-4, N0-3b, M0 mucosal head and neck squamous cell carcinoma (HNSCC) (oral cavity, oropharynx, larynx, hypopharynx, nasal cavity and skin) that is appropriate for surgical resection. Both previously untreated (primary) and recurrent (salvage) settings will be eligible. Tumors must be accessible to biopsy in clinic (patients with laryngeal, hypopharyngeal, nasal cavity and base of tongue tumors will have endoscopic biopsies)

• For patients with oropharyngeal cancer, only p16-negative (non-human papillomavirus \[HPV\] related) patients will be eligible

• Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray or as ≥ 10 mm (≥ 1 cm) with CT scan, MRI, or calipers by clinical exam

• Age ≥ 18 years. Because no dosing or adverse event (AE) data are currently available on the use of cemiplimab (REGN2810) alone or in combination with CDX-1140 in patients \< 18 years of age, children are excluded from this study

• Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 (Karnofsky ≥ 60%)

• Hemoglobin (Hb) ≥ 7 g/dL (transfusion allowed to bring Hb to this level)

• Absolute neutrophil count ≥ 1,000/mcL

• Platelets ≥ 100,000/mcL

• Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN)

• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) /alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 × institutional ULN

• Creatinine ≤ 1.5 × institutional ULN OR glomerular filtration rate (GFR) ≥ 60 mL/min/1.73 m\^2

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated

• Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load

• Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better

• Based on its mechanism of action, cemiplimab (REGN2810) can cause fetal harm when administered to a pregnant woman. Animal studies have demonstrated that inhibition of the PD-1/PD-L1 pathway can lead to increased risk of immune-mediated rejection of the developing fetus resulting in fetal death. Women of childbearing potential and men should use effective contraception during treatment with cemiplimab (REGN2810) and for 4 months after the last dose. The reproductive and developmental toxicity of CDX-1140 has not been evaluated. Women of childbearing potential and their partners who receive CDX-1140 must therefore take adequate contraceptive measures

• Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants