Inclusion Criteria:  Meets FDA-approved criteria for treatment of non-Hodgkin lymphoma (NHL) with axicabtagene ciloleucel (Yescarta), tisagenlecleucel (Kymriah), lisocabtagene maraleucel (Breyanzi) or brexucabtagene autoleucel (Tecartus). Subjects receiving breuxacabtagene autoleucel for treatment of B-cell acute lymphoblastic leukemia (ALL) are not eligible. At least 18 years of age. The effects of duvelisib on the developing human fetus are unknown. For this reason, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry, for the duration of study participation, and for at least 3 months after the last dose of duvelisib, as well as conform to institutional CAR T-cell guidelines. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must 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 the study, and for at least 3 months after the last dose of duvelisib. Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable).  Exclusion Criteria:  Receiving axicabtagene ciloleucel, tisagenlecleucel, lisocabtagene maraleucel, or brexucabtagene autoleucel for the treatment of B-cell acute lymphoblastic leukemia. Known allergy or intolerance to duvelisib or another PI3K inhibitor. Previous treatment with duvelisib or other PI3K inhibitor is permitted unless therapy was discontinued due to toxicity or intolerance of therapy. Receiving therapy with a strong CYP3A inducer or inhibitor that cannot be discontinued during duvelisib therapy. Subjects receiving a strong CYP3A inducer or inhibitor at screening are eligible to participate if the drug can be discontinued the longest of the following time periods prior to initiation of duvelisib: 7 days (for strong CYP3A inhibitors), 14 days (for strong CYP3A inducers) or 4-5 half-lives (either inducer or inhibitor). Active CNS involvement by hematologic malignancy under treatment Evidence of uncontrolled infection of any origin (viral, bacterial, or fungal) Active bacterial, fungal or mycobacterial infection tuberculosis requiring treatment within the two years prior to study enrollment Known HIV infection, untreated hepatitis C or hepatitis B infection. Untreated hepatitis B is not an exclusion if hepatitis B is undetectable. Acute or chronic GVHD requiring systemic therapy Concurrent use of chronic systemic steroids or immunosuppressant medications Known history of immunologic/autoimmune disease affecting the CNS unrelated to diagnosis of hematologic malignancy under treatment Clinically significant venous thromboembolic disease, defined deep vein thrombosis or pulmonary embolism occurring in the last 3 months or requiring ongoing anticoagulation at time of study screening Clinically significant pulmonary disease, defined as grade 2 or greater dyspnea or grade 2 or greater hypoxia Clinically significant cardiac disease, defined as unstable angina, acute myocardial infarction in the last 6 months, and NYHA class II or IV heart failure. Subjects with unstable arrhythmias that are not stable with medical management in 2 weeks prior to day -2 are also excluded. Clinically significant hepatic disease, defined as ALT, AST or alkaline phosphatase ≥ 3x ULN or total bilirubin > 1.5x ULN (unless related to Gilbert's or Meulengracht's syndrome). Subjects with a history of chronic liver disease, previous veno-occlusive disease, active alcohol abuse or history of alcohol abuse within the past 6 months are also excluded. Clinically significant renal disease, defined as calculated or measured creatinine clearance < 50 mL/min Currently breastfeeding or pregnant. Women of childbearing potential must have a negative pregnancy test within 7 days of study entry. Inability to swallow and retain oral medication or prior surgery or GI dysfunction that may affect drug absorption (i.e. gastric bypass surgery, gastrectomy) Receipt of a prior investigational agent within 4 weeks before Day -3 or currently receiving any other investigational agents. Unable to receive prophylactic treatment for pneumocystis, HSV or VZV at screening Any condition that would, in the investigator's judgment, interfere with full participation in the study, including administration of medication, attendance of study visits, elevated risk of complications or interference with interpretation of the study data A history of other malignancy with the exception of malignancies for which all treatment was completed at least 2 years before registration and the patient has no evidence of disease. Non-metastatic, non-melanoma skin cancers are not considered exclusionary.