Testing the Addition of an Anti-Cancer Drug, Irinotecan, to the Standard Chemotherapy Treatment (FOLFOX) After Long-Course Radiation Therapy for Advanced-Stage Rectal Cancers to Improve the Rate of Complete Response and Long-Term Rates of Organ Preservation

Inclusion Criteria: * Stage: Clinical stage II or III rectal adenocarcinoma defined as T4N0 or any T with node positive disease (any T, N+); also T3N0 requiring abdominal perineal resection (APR) or coloanal anastomosis * Tumor site: Rectum; =\< 12cm from the anal verge * No prior systemic chemotherapy, targeted therapy, or immunotherapy; or radiation therapy administered as treatment for colorectal cancer within the past 5 years is allowed * Not pregnant and not nursing, because this study involves an agent that has known genotoxic, mutagenic and teratogenic effects \* Therefore, for women of childbearing potential only, a negative pregnancy test (urine or serum according to institutional guidelines) done =\< 14 days prior to registration is required. Female subjects agree to use highly effective contraception combined with an additional barrier method (e.g, diaphragm, with a spermicide) while on study and for \>= 9 months after last dose of study drug, and the same criteria are applicable to male subjects if they have a partner of childbirth potential. Male subject agrees to use a condom and not donate sperm while in this study and for \>= 6 months after the last treatment * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (or Karnofsky \>= 60%) * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm * Creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated (calc.) creatinine clearance \>= 50 mL/min …