Modified Immune Cells (CD19-CD22 CAR T Cells) in Treating Patients With Recurrent or Refractory CD19 Positive, CD22 Positive Leukemia or Lymphoma

Inclusion Criteria:

* Patients with relapsed/refractory B-acute lymphoblastic leukemia (ALL), chronic lymphocytic leukemia (CLL), or non-Hodgkin's lymphoma (NHL) treated with at least two lines of therapy, and have persistent or progressed disease including positive minimal residual disease (MRD)
* Patients may have received last cytotoxic chemotherapy at least 3 weeks prior to lymphodepleting chemotherapy
* Patient may continue targeted therapy until 2 weeks before initiation of lymphodepleting chemotherapy with the exception of ibrutinib
* Disease must be CD19 and/or CD22 positive by flow cytometry or immunohistochemistry
* Karnofsky/Lansky performance scale \> 70
* Total bilirubin less than \< 1.5 mg/dL except patients with Gilbert syndrome whose total bilirubin must be \< 3.0mg/dL
* Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\]) =\< 2.5 X upper limit of normal (ULN)
* Alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 5.0 ULN
* Serum creatinine (as estimated by Cockcroft Gault) \>= 60 cc/min
* Cardiac ejection fraction \>= 50% without evidence of pericardiac effusion as determined by echocardiogram (ECHO) or multigated acquisition scan (MUGA), no clinical significant electrocardiogram (ECG) findings
* No clinical significant pleural effusion and baseline oxygen saturation \>= 92%
* Absolute lymphocyte count \>= 100/ul
* Be able to sign informed consent
* All participants who are able to have children must practice effective birth control while on study. Acceptable forms of birth control for female patients include: birth control pills, patches, or injections, intrauterine device (IUD), diaphragm with spermicide, or condom with spermicide. Acceptable forms of birth control for male patients include condom with spermicide. If female participant becomes pregnant during the study, she will be taken off this study. If male participant fathers a child while on study, he must immediately notify his doctor
* For patients with history of allogenic stem cell transplantation

  * Should not have active acute graft-versus-host disease (GVHD) grade \>= 2
  * Should not be on immunosuppressants such as tacrolimus, sirolimus, cyclosporine, mycophenolates for a minimum of a month from CD19-CD22 CAR T cell infusion
  * Should not be on more than physiologic dose of systemic steroid for adrenal insufficiency (prednisone equivalent 5mg/day)
  * Transplantation should be more than 2 months from CD19-CD22 CAR T cell infusion
  * Other cell therapy including CAR T cells, donor lymphocyte infusion, virus specific T cells, natural killer (NK) cells, etc should have 6 weeks of wash-out period from the CD19-CD22 CAR T cell infusion
* For patients with history of central nervous system (CNS) disease, CNS disease must be treated prior to enrollment
* For patients with prior treatment history of cell therapy such as other CAR T cells or CAR NK cells or NK cells, cell therapy should have a 6 weeks of wash-out period from CD19-CD22 CAR T cell infusion
* Be able to consent long-term follow-up protocol PA17-0483

Exclusion Criteria:

* Positive beta-human chorionic gonadotropin (hCG) in female of child bearing potential defined as not postmenopausal for 24 months or no previous surgical sterilization or lactating females
* Known positive serology for human immunodeficiency virus (HIV)
* Presence of active grade 3 or greater toxicity from the previous treatment
* Presence of active fungal, bacterial, viral, or other infection requiring IV antibiotics for management
* Presence of active neurologic disorders
* Concomitant use of other investigational agents
* Current use of corticosteroid more than physiological dose for adrenal insufficiency (prednisone equivalent at a dose higher than 10 mg/day)
* Presence of active CNS disease