Brief Summary: This phase II trial studies how well trifluridine/tipiracil and irinotecan work in treating patients with biliary tract cancer that has spread to other places in the body (advanced) and has not responded to treatment (refractory). Trifluridine/tipiracil and irinotecan may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth.

Detailed Description: PRIMARY OBJECTIVE: I. Determine the efficacy of trifluridine and tipiracil hydrochloride (trifluridine/tipiracil) in combination with irinotecan hydrochloride (irinotecan) in patients with refractory biliary tract cancers using progression-free survival (PFS) at 16 weeks. SECONDARY OBJECTIVES: I. Assess the safety and tolerability of trifluridine/tipiracil in combination with irinotecan in patients with refractory biliary tract cancers through adverse event monitoring. II. Further explore the efficacy of trifluridine/tipiracil in combination with irinotecan in patients with refractory biliary tract cancers by overall response rates (ORR), disease control rates (DCR), and overall survival (OS). CORRELATIVE RESEARCH: I. To determine if the number of circulating tumor cells (CTCs) or the level of cell-free deoxyribonucleic acid (DNA) (cfDNA) at baseline is prognostic or predictive to the response to therapy. II. To determine if changes in CTCs or cfDNA correlate with efficacy endpoints. III. To determine if drug response from a parallel ex vivo trial using patient-derived tumor organoid correlates with clinical response to trifluridine/tipiracil plus irinotecan. IV. To evaluate the role of thymidine kinase 1 (TK1) in predicting the clinical benefit of trifluridine/tipiracil plus irinotecan and discover potential mechanisms of resistance using patient-derived tumor organoid and pre-treatment biopsy specimen. EXPLORATORY RESEARCH: I. To evaluate patients who received prior treatment with fluorouracil (5-FU) independently from the entire population in the following areas: PFS, safety and tolerability, ORR, DCR, and OS. OUTLINE: Patients receive trifluridine and tipiracil hydrochloride orally (PO) twice daily (BID) on days 1-5 and irinotecan hydrochloride (IV) over 90 minutes on day 1. Cycles repeat every 14 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up at 30 days, then every 3 months for up to 2 years after study registration.