Viral Therapy in Treating Patient With Refractory Liver Cancer or Advanced Solid Tumors

Inclusion Criteria: * ARM A: Histologically or cytologically confirmed hepatocellular carcinoma that is refractory (by Response Evaluation Criteria in Solid Tumors \[RECIST\] or modified \[m\]RECIST criteria or with unequivocal clinical progression of disease) to or intolerant (defined as inability to administer further sorafenib due to drug related toxicities) of sorafenib based therapy or advanced solid tumor with liver predominant disease burden that has progressed on or is intolerant to standard * ARM A: Absolute neutrophil count (ANC) \>= 1000/mm\^3 * ARM A: Platelet count \>= 80,000/mm\^3 * ARM A: Hemoglobin \>= 10 g/dl * ARM A: Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 5 x ULN * ARM A: Creatinine =\< 1.5 x ULN * ARM A: Total bilirubin =\< 1.5 x ULN * ARM A: International normalized ratio (INR) =\< 1.5 x ULN * ARM A: Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN * ARM A: Ability to provide informed written consent * ARM A: Willingness to return to Mayo Clinic in Arizona for follow-up * ARM A: Life expectancy \>= 12 weeks * ARM A: Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * ARM A: Willingness to provide all biological specimens as required by the protocol * ARM A: Negative serum pregnancy test =\< 7 days prior to registration for women of childbearing potential only * ARM A: Child Pugh Score A or B7 (patients with ascites must have paracentesis performed within scope of standard of care, to be able to…