Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed locally recurrent HNSCC
             for whom re-irradiation for local control is considered standard of care

          -  Patients with human papillomavirus (HPV)-negative or HPV-positive head and neck cancer
             are eligible

          -  Patients who have had prior treatment with immune therapies are eligible

          -  Patients must have received curative-intent platinum- and/or cetuximab-based
             chemoradiotherapy or radiotherapy alone

          -  Patients must have completed their last treatment dose with chemotherapy or
             immunotherapy at least 4 weeks (6 weeks for nitrosoureas or mitomycin C) before
             enrolling on study

          -  Patients must have completed their last treatment dose with radiotherapy at least 6
             months before enrolling on study

          -  Patients who have had major surgery must be fully recovered and require a recovery
             period of at least 4 weeks prior to enrolling on study

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2

          -  Hemoglobin >= 9 g/dL (transfusion permitted)

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Total bilirubin within 1.5 x the upper limit of normal (ULN) institutional limits

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional upper limit of normal

          -  Creatinine clearance >= 50 mL/min/1.73 m^2 for patients with creatinine levels above
             institutional normal. Creatinine clearance will be determined using the Chronic Kidney
             Disease Epidemiology Collaboration (CKD-EPI) equation

          -  Patients must have a corrected QT interval by Fredericia (QTcF) =< 480 msec

          -  International normalized ratio (INR) =< 1.5 and no clinically significant bleeding
             event within the past six months

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Patients must have measurable disease, per Response Evaluation Criteria in Solid
             Tumors (RECIST) 1.1

          -  The effects of birinapant on the developing human fetus are unknown. For this reason,
             women of child-bearing potential and men must agree to use adequate contraception
             (hormonal or barrier method of birth control; abstinence) beginning at study entry and
             for the duration of study participation. Male study participants should use an
             additional barrier method of contraception for 30 days following the last dose of
             birinapant. 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

        Exclusion Criteria:

          -  Eligibility for curative-intent surgery

          -  More than 2 lines of palliative systemic therapy (platinum-, taxane- or
             cetuximab-based chemotherapy or immunotherapy)

          -  Patients who are receiving any other investigational agents

          -  Patients with known brain metastases should be excluded from this clinical trial
             because of their poor prognosis and because they often develop progressive neurologic
             dysfunction that would confound the evaluation of neurologic and other adverse events

          -  Participants with nasopharyngeal carcinoma are excluded

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to birinapant

          -  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 because birinapant may have potential for
             teratogenic or abortifacient effects. Because there is an unknown but potential risk
             for adverse events in nursing infants secondary to treatment of the mother with
             birinapant, breastfeeding should be discontinued prior to enrollment. A negative
             pregnancy test is required for women of childbearing potential. Women who are
             postmenopausal (age-related amenorrhea >= 12 consecutive months, or who had undergone
             hysterectomy or bilateral oophorectomy are exempt from pregnancy testing. If
             necessary, to confirm postmenopausal status, a follicle stimulating hormone (FSH)
             level may be included at screening

          -  Human immunodeficiency virus (HIV) positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             birinapant

          -  Patients requiring the use of anti-tumor necrosis factor (anti TNF) therapies, such as
             infliximab, or patients who have received treatment with anti-TNF therapies within 5
             half-lives of the drug (48 days for infliximab, 55 days for golimumab, 70 days for
             certolizumab and adalimumab, and 16 days for etanercept)

          -  Patients with previous exposure to birinapant