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 eithe…