Bispecific CD19/CD20-Targeted Chimeric Antigen Receptor (CAR) Modified T Cells With 4-1BB and Mutated CD28 Costimulatory Domains in Patients With Relapsed or Refractory CD19+ Hematologic Malignancies

Inclusion Criteria: * To be eligible for leukapheresis, patients must have an aggressive CD19+ B cell malignancy with relapsed or refractory disease, defined as below. * To be eligible for treatment with 20-19 Tan BB06z CAR-T cells, patients must additionally have detectable evidence of residual malignancy at the time of assessment prior to CAR-T cell infusion, regardless of therapy administered following leukapheresis. To be included in this study, participants must meet the following criteria: 1\. Patients with R/R B-Cell malignancies (see below) which commonly express CD19 and CD20. Eligible disease subtypes include the following: o DLBCL/HGBL de novo or DLBCL/HGBL transformed from an indolent lymphoma, refractory to front-line chemoimmunotherapy containing an anthracycline and CD20-directed therapy (without achievement of CR). * Relapsed or refractory DLBCL/HGBL following 2 or more prior chemoimmunotherapy regimens containing an anthracycline and CD20-directed therapy following diagnosis of de novo DLBCL/HGBL or DLBCL arising from indolent lymphoma. * Relapse following a single prior chemoimmunotherapy regimen containing an anthracycline and CD20-directed therapy following diagnosis of de novo DLBCL/HGBL or DLBCL arising from indolent lymphoma and considered ineligible for high dose chemotherapy and autologous stem cell rescue as determined by the investigator. * Mantle Cell Lymphoma after 2 lines of therapy. Patients must have previously received chemoimmunotherapy and a prior BTK inhibitor. * Secondary CNS Lymphoma after 2 lines of therapy, 1 of which must include an autologous stem cell transplant or deemed ineligible by the investigator. * Patients with B cell acute lymphoblastic leukemia (ALL) and chronic myeloid leukemia (CML) in lymphoid blast crisis. * Patients with Philadelphia chromosome-negative B cell ALL must have been refractory to at least 1 line of multi-agent chemotherapy or relapsed following at least 1 prior multiagent systemic chemotherapy regimen that included induction and consolidation therapy. * Patients with Philadelphia chromosome-positive ALL or CML in lymphoid blast crisis must have exhibited persistent disease following therapy with a second- or third-generation tyrosine kinase inhibitor. * Burkitt lymphoma refractory to at least 1 line of multi-agent chemotherapy or relapsed following 1 or more lines of multi-agent chemotherapy. * Patients must have at least one FDG-avid (PET-avid) measurable lesion. • For B-cell ALL and/or CML in lymphoid blast crisis, presence of \> 20% lymphoid blasts in the blood or bone marrow, or extramedullary infiltration with FDG-avid (PET-avid) measurable lesion is required. 3\. Prior CD19-targeted therapies, including CAR-T cell therapy, do not exclude participation. However, CD19 and CD20 expression by immunohistochemical staining or flow cytometry must be confirmed prior to enrollment for patients who have received such therapies. * Have the following clinical laboratory values: * Adequate renal function defined as; \- Adult participants: • Creatinine Clearance \>30 mL/min (Cockroft-Gault equation) \- Pediatric participants: Use age/gender -appropriate creatinine as follows: Age Maximum Plasma Creatinine (mg/dL) Male Female 10 to \< 13 years 1.2 1.2 13 to \< 16 years 1.5 1.4 ≥ 16 years 1.7 1.4 * Direct bilirubin ≤2.0 mg/dL * AST and ALT ≤3.0x upper limit of normal (ULN) * Adequate pulmonary function as assessed by ≥92% oxygen saturation on room air by pulse oximetry. * ECOG performance status 0-1 for adult participants ≥18 years of age. Pediatrics - Lansky/Karnofsky score ≥70: use Karnofsky for participants ≥16 years of age and Lansky for participants \<16 years of age (see Appendix A). * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry. 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. Participants of childbearing age should use effective contraception while on this study and continue for 1 year after all treatment is finished. 9\. Participant, or legal representative, must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure. Exclusion Criteria: * Pregnant or lactating patients. * Impaired cardiac function (LVEF \<40%) as assessed by ECHO or MUGA scan during screening. * Patients with active known autoimmune disease requiring systemic T cell suppressive therapy are ineligible. * Patients with active graft versus host disease following allogeneic hematopoietic cell transplantation requiring systemic T cell suppressive therapy are ineligible. * Patients with following cardiac conditions will be excluded: * New York Heart Association (NYHA) stage III or IV congestive heart failure * Myocardial infarction ≤6 months prior to enrollment * Any history of clinically significant ventricular arrhythmia or unexplained syncope, not believed to be vasovagal in nature or due to dehydration * Patients with HIV are ineligible. * Patients with active hepatitis B infection (as manifest by either detectable hepatitis B virus DNA by PCR and/or positivity for hepatitis B surface antigen) are ineligible. * Patients with active hepatitis C infection (as manifest by detectable hepatitis C virus RNA by PCR) are ineligible. Patients with detectable antibodies to hepatitis C virus will be screened by PCR for evidence of active infection. * Patients with uncontrolled systemic fungal, bacterial, viral or other infection are ineligible. * Patients with any concurrent active malignancies as defined by malignancies requiring any therapy other than expectant observation or hormonal therapy, with the exception of squamous and basal cell carcinoma of skin. * Patients with history or presence of clinically significant neurological disorders such as epilepsy, generalized seizure disorder, severe brain injuries are ineligible. * Unwilling or unable to follow protocol requirements. * Any other condition/issue which, in the opinion of the treating physician, would make the patient ineligible for the study.