Ponatinib Hydrochloride in Treating Patients With Advanced Biliary Cancer With FGFR2 Fusions

Inclusion Criteria: * Histological/cytological confirmation of biliary cancer * Confirmation of advanced biliary cancer that is refractory or intolerant to gemcitabine or fluoropyrimidine based therapy with FGFR2 fusion \[using next-gen sequencing assays (such as Foundation One) or fluorescent in situ hybridization (FISH) break-apart assays\] or FGFR pathway mutation/amplification \[using next-gen sequencing assays (such as Foundation One)\]; assays must be performed in a Clinical Laboratory Improvement Amendments \[CLIA\] certified laboratory and done as a CLIA validated test or research use only \[RUO\] in a CLIA laboratory * Measurable disease * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Hemoglobin \>= 9.0 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome * Aspartate transaminase (AST) and alanine aminotransferase (ALT) \< 3 x ULN * Creatinine =\< 1.5 x ULN * Serum lipase and amylase =\< 2.5 x ULN; NOTE: if subject has tumor involvement in the liver =\< 3 x ULN * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Recovered from prior radiotherapy and/or systemic therapy related toxicities to grade =\< 1 * Provide informed written consent * Life expectancy \>= 3 months * Willing to return to enrolling institution for follow-up (during the Active Monitoring P…