Ponatinib for Patients Whose Advanced Solid Tumor Cancer Has Activating Mutations Involving the Following Genes: FGFR1, FGFR2, FGFR3, FGFR4, RET, KIT.

Inclusion Criteria: * Patients with histologically or cytologically confirmed diagnosis of refractory metastatic solid tumor or chronic hematological malignancy who are eligible for investigational drug therapy * Patients must have tumor suitable for biopsy (as assessed by trained specialists in interventional radiology) and medically fit to undergo a biopsy or surgical procedure OR if patients do not have a tumor suitable for biopsy but have another tissue available for molecular evaluation * Patients should have activating genomic alterations in FGFR (mutations, fusions or amplifications \[\> 6 copies\]) or activating genomic alterations in KIT, platelet-derived growth factor receptor alpha \[PDGFRα\], ret proto-oncogene \[RET\], ABL proto-oncogene 1, non-receptor tyrosine kinase \[ABL1\] and fms-related tyrosine kinase 3 \[FLT3\] by any validated Clinical Laboratory Improvement Amendments \[CLIA\]-certified molecular testing (fluorescent in situ hybridization \[FISH\], polymerase chain reaction \[PCR\] or sequencing data are acceptable); CLIA validated results from other institutions; diagnostic labs (e.g. foundation medicine) are acceptable; additional types of activating alterations in these genes can be approved by the principal investigator (PI) * Patients with advanced cancers should have had at least one prior therapy that is considered standard for that disease type * Patients with solid tumors must have measurable disease (Response Evaluation Criteria in Solid Tumors \[RECIST\] 1.1), defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 80%) * Life expectancy of greater than 3 months * Patients with multiple malignancies remain eligible * Patients with an inherited cancer syndrome or a medical history suggestive of an inherited cancer syndrome remain eligible * Patients must have controlled blood pressure with a systolic blood pressure \< 140 mmHg and diastolic \< 90 mmHg; anti-hypertensive medications are permitted * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and through 4 months after the end of treatment; for females of childbearing potential, a negative pregnancy test must be documented prior to randomization * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 75,000/mcL * Total bilirubin =\< 1.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome (\< 5 if liver involvement with primary tumor) * Serum lipase and amylase =\< 1.5 x ULN * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Left ventricular ejection fraction (LVEF) \>= institutional lower limit of normal by echocardiogram (ECHO) or multi gated acquisition (MUGA) * Serum creatinine =\< 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault formula) \>= 50 mL/min OR 24-hour urine creatinine clearance \>= 50 mL/min * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients with acute hematological malignancies * Patients who have not received any prior treatment. * Patients with known ponatinib-resistant gene alterations * PDGFRA D842V mutation * cKIT D816V mutation * FLT3 D835V/Y/H/F or Y842C mutations * FGFR3 K652E mutation * Major surgery (e.g. thoracic, abdominal, vascular, neurosurgery) within 28 days prior to initiating therapy * History of acute pancreatitis within one year of study or history of chronic pancreatitis * History of alcohol abuse * Have uncontrolled hypertriglyceridemia (triglycerides \> 450 mg/dL) * Patients with history of clinically significant bleeding disorder * Pregnant women are excluded from this study because ponatinib can affect embryo-fetal development. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with ponatinib breastfeeding must be discontinued. * Patients who are incarcerated are not eligible * Patients with any history of arterial thromboembolic disease; any patient with a history of myocardial infarction (MI), stroke, transient ischemic attack (TIA), unstable angina or peripheral vascular disease will not be eligible * Patients with history of recurrent venous thromboembolism (deep venous thrombosis or pulmonary embolism) or history of venous thromboembolism within 6 months will not be eligible * Patients with history of active hepatitis B or C infection or chronic hepatitis with Child Pugh B or C hepatic dysfunction * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ponatinib * Patients with prolonged corrected QT interval, defined as QTc \>450 msec * Use of antiplatelet agents other than low-dose aspirin as described * GI bleed within 30 days prior to registration on study * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ponatinib. * Patients with history of atrial arrhythmia (requiring any anti-arrhythmic therapy) or patients with any history of ventricular arrhythmia are excluded * Clinically significant, uncontrolled intercurrent illness including, but not limited to: * Symptomatic or active infection * Uncontrolled hypertension (diastolic blood pressure \> 90 mm Hg; systolic \> 140 mm Hg); patients with hypertension should be under treatment on study entry to effect blood pressure control * Psychiatric illness/social situations that would limit compliance with study requirements * Patients with history of congestive heart failure are excluded * HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with ponatinib. * Patients on medications known to be associated with Torsades de Pointes * Patients who received the last administration of an anti-cancer therapy including, chemotherapy, immunotherapy/biologic therapy, targeted therapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) or within 5 half-lives, whichever is shorter, prior to entering the study. * Patients taking medications or herbal supplements that are known to be strong cytochrome P450 3A4 (CYP3A4) inhibitors within at least 14 days before the first dose of ponatinib are excluded * Patients with symptomatic or progressive brain metastases are ineligible; subjects with treated brain metastases are eligible if they have no radiographic or other signs of progression in the brain for \>= 4 weeks after completion of local therapy * Patients with macular edema, retinal vein occlusion or retinal hemorrhage are excluded. * Patients who have received prior FGFR targeted therapy