Lapatinib in Treating Patients With Locally Advanced or Metastatic Biliary Tract or Liver Cancer That Cannot Be Removed By Surgery

Inclusion Criteria: * Patients must have histologically or cytologically confirmed, surgically unresectable biliary cancer (gallbladder, ampullary, intra or extrahepatic bile duct) OR patients must have surgically unresectable HCC and who are not candidates for percutaneous ethanol injection or radio frequency ablation (RFA); patients must have histological or cytological confirmation of HCC * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral CT scan; if a patient has undergone TACE, ethanol or RFA ablation then new lesions need to be present in the liver, if there are no other sites of disease * Patients may have received prior therapy as follows: * No more than one prior chemotherapy regimen for metastatic or recurrent disease will be allowed; prior chemotherapy for earlier stage disease (neoadjuvant, adjuvant, or concurrent with radiation therapy) will be allowed in addition to prior chemotherapy for recurrent metastatic disease; TACE is considered one regimen; at least 3 weeks must have elapsed since prior therapy, and toxicities of therapy should have resolved to =\< Grade 1 * Patients may have received prior radiation therapy; three weeks must have elapsed since the completion of prior radiation therapy and patients must have recovered from all toxicities; measurable disease must either be outside the previous radiation field, or progressing within a radiated field, or a new lesion must be present * There must be no plans for the patient to receive concurrent hormonal, biologic, or radiation therapy to measurable lesions * Patients who have had prior treatment with EGFR targeting therapies are ineligible * Patients may not be receiving any other investigational agents or receiving concurrent anticancer therapy * Life expectancy of greater than 12 weeks * ECOG performance status less than or equal to 2 (Karnofsky \>= 60%) * Patients who have scores that fall into Childs B or C groups are excluded * Patients must have organ and marrow function as defined below: * Leukocytes \>= 3000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 75,000/mcL * Total bilirubin \< 2 mg/dl * AST(SGOT)/ALT(SGPT) =\< 5.0 X upper limit of institutional normal (ULN) * PT prolongation \< 4 secs above ULN (unless taking warfarin) * Creatinine =\< ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Cardiac ejection fraction above the lower limit of normal as measured by echocardiogram or MUGA scan; note that baseline and on- treatment scans should be performed using the same modality and preferably at the same institution * Adherence to the requirements for concomitant medications classified as CYP3A4 inducers or inhibitors * HIV-positive patients receiving combination anti-retroviral therapy are excluded from the study because of possible pharmacokinetic interactions with GW572016; appropriate studies will be undertaken in patients receiving combination anti-retroviral therapy when indicated * Patients requiring oral anticoagulants (coumadin, warfarin) are eligible provided there is appropriate close INR monitoring is in place; if medically appropriate and treatment available, the investigator may also consider switching these patients to low molecular weight (LMW) heparin, where an interaction with GW572016 is not expected * The effects of GW572016 on the developing human fetus at the recommended therapeutic dose are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control or abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Pregnant women are excluded from this study because GW572016 is member of the 4-anilinoquinazoline class of kinase inhibitors with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with GW572016, breastfeeding should be discontinued if the mother is treated with GW572016 * Ability to understand and the willingness to sign a written informed consent document * Able to swallow and retain oral medication * Patients with known brain metastases (Scans are not necessary to exclude brain metastasis) should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * Patients with 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, will be excluded * Patients with GI tract disease resulting in an inability to take oral medication, malabsorption syndrome, a requirement for IV alimentation, prior surgical procedures affecting absorption, uncontrolled inflammatory GI disease (e.g., Crohn's, ulcerative colitis) are also ineligible * History of allergic reactions attributed to compounds of similar chemical or biologic composition to GW572016 (i.e inhibitors of EGF and HER2- /neu) will render patients ineligible