Trifluridine/Tipiracil and Irinotecan for the Treatment of Advanced Refractory Biliary Tract Cancer

Inclusion Criteria: * Histological confirmation of advanced biliary tract cancers including cancers originating in the gallbladder who have received at least one line of systemic anticancer therapy * Note: Patients who have either progressed on or are intolerant to the prior therapy can be included in this study * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) criteria * NOTE: Tumor lesions in a previously irradiated area are not considered measurable disease. Disease that is measurable by physical examination only is not eligible * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0 or 1 * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (=\< 21 days prior to registration) * Platelet count \>= 100,000/mm\^3 (=\< 21 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (=\< 21 days prior to registration) * Aspartate transaminase (AST) or alanine transaminase (ALT) =\< 3 x ULN (=\< 21 days prior to registration) * Creatinine =\< 1.5 x ULN (=\< 21 days prior to registration) * Negative pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * Provide written informed consent * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * Willingness to provide mandatory blood and tissue specimens for correlative research Exclusion Criteria: * Any of the following because this study involves an agent that has potential genotoxic, mutagenic and teratogenic effects: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ adequate contraception for at least 3 months after the last dose of the study drug * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy * NOTE: Patients known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm =\< 21 days prior to registration * Receiving any anticancer therapy for biliary tract cancer =\< 21 days prior to registration * Other active malignancy requiring treatment in =\< 6 months prior to registration * EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix * NOTE: If there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * History of myocardial infarction =\< 6 months prior to registration, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Previous treatment with irinotecan or irinotecan-based chemotherapy for biliary tract cancers