Dasatinib in Treating Patients With Advanced Liver Cancer That Cannot Be Removed by Surgery

Criteria: * WBC \>= 3,000/mm\^3 * LVEF normal * Histologically or cytologically confirmed hepatocellular carcinoma; Advanced disease, unresectable disease, no Childs C criteria * Measurable disease, defined as \>= 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques or \>= 10 mm by spiral CT scan * Not a candidate for percutaneous ethanol injection or radiofrequency ablation (RFA) * Prior transarterial chemoembolization, ethanol, and RFA allowed if new lesions are present in the liver and there are no other sites of disease * No pleural effusion or ascites requiring paracentesis within the past 4 weeks * No known brain metastases * ECOG performance status (PS) 0-1 OR Karnofsky PS 70-100% * Life expectancy \> 3 months * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 75,000/mm\^3 * Bilirubin =\< 2 times upper limit of normal (ULN) * AST and ALT =\<2.5 times ULN (5 times ULN if liver involvement by tumor) * Creatinine =\< 2 times ULN * PT =\< 1.5 times ULN (no anticoagulation) * Albumin \>= 2.5 mg/dL * No history of allergic reactions attributed to compounds of similar chemical or biological composition to dasatinib * No evidence of encephalopathy * No condition that would preclude ability to swallow and retain dasatinib tablets, including any of the following: * Gastrointestinal tract disease resulting in an inability to take oral medication; * Requirement for IV alimentation; * Prior surgical procedures affecting absorption: * Active peptic ulcer disease * No clinically significant ECG abnormalities * No clinically significant cardiovascular disease, including any of the following: * Myocardial infarction or ventricular tachyarrhythmia within the past 6 months; * Prolonged QTc \>= 480 msec (Fridericia correction); * Major conduction abnormality (unless cardiac pacemaker is present) * No other uncontrolled illness, including, but not limited to, any of the following: * Ongoing or active infection; * History of significant bleeding disorder, including congenital (von Willebrand's disease) or acquired disorders (antifactor VIII antibodies); * Psychiatric illness or social situation that would preclude study compliance * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * Recovered from all prior therapy * One prior systemic chemotherapy regimen allowed * Prior cryosurgery allowed * More than 4 weeks since prior transarterial chemoembolization * More than 4 weeks since prior radiotherapy * Prior or concurrent localized palliative radiotherapy (i.e., bony metastasis) allowed provided it was administered for =\< 3 days * At least 7 days since prior and no concurrent antithrombotic and/or antiplatelet agents (e.g., warfarin, heparin, low molecular weight heparin, acetylsalicylic acid, and/or ibuprofen) * At least 7 days since prior and no concurrent agents with proarrhythmic potential * At least 7 days since prior and no concurrent medications or substances that are potent inhibitors or inducers of CYP3A4 * No concurrent combination antiretroviral therapy for HIV-positive patients * No concurrent embolization or chemoembolization * No concurrent systemic antacids (H2 receptor antagonists or proton pump inhibitors) * Locally active antacids allowed provided they are held for 2 hours before and 2 hours after dasatinib dose * No other concurrent investigational agents * No other concurrent anticancer agents or therapies