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 successfully perform intratumoral injection procedure) * ARM A: The patient and their partner agree to use a barrier method of contraception during the study and 4 months following end of active treatment * ARM A: Disease burden in liver not affecting more than 25% of liver * ARM A: Predominant intrahepatic burden (\> 75%) of disease (i.e. patients with widespread extrahepatic disease to organs other than the liver will not be included) * ARM B: Histologically or cytologically confirmed solid tumor with subcutaneous/cutaneous lesions that is refractory (RECIST or with unequivocal clinical progression of disease) to or intolerant to standard therapy * ARM B: Absolute neutrophil count (ANC) \>= 1000/mm\^3 * ARM B: Platelet count \>= 100,000/mm\^3 * ARM B: Hemoglobin \>= 10 g/dl * ARM B: AST/ALT =\< 2.5 x ULN * ARM B: Creatinine =\< 1.5 x ULN * ARM B: Total bilirubin =\< 1.5 x ULN * ARM B: INR =\< 1.5 x ULN * ARM B: aPTT =\< 1.5 x ULN * ARM B: Ability to provide informed written consent * ARM B: Willingness to return to Mayo Clinic in Arizona for follow-up * ARM B: Life expectancy \>= 12 weeks * ARM B: ECOG performance status (PS) 0 or 1 * ARM B: Willingness to provide all biological specimens as required by the protocol * ARM B: Negative serum pregnancy test =\< 7 days prior to registration for women of childbearing potential only * ARM B: Child Pugh Score A * ARM B: The patient and their partner agree to use a barrier method of contraception during the study and 4 months following end of active treatment * ARM B: Disease burden in liver not affecting more than 25% of liver Exclusion Criteria: * ARM A: Uncontrolled infection * ARM A: Systemic anti-cancer therapy =\< 4 weeks prior to registration * ARM A: Known human immunodeficiency virus (HIV) infection * ARM A: Other concurrent chemotherapy, immunotherapy, radiotherapy, or investigational therapy * ARM A: Pregnant or nursing women * ARM A: History of bone marrow or solid organ transplantation * ARM A: Patient for whom surgical resection or liver transplantation would be more appropriate * ARM A: Any condition, which in the opinion of the investigator would render the patient unsuitable to participate in the study * ARM A: Any corticosteroid use =\< 28 days prior to registration * ARM A: Any radioembolization or transarterial chemoembolization (TACE) =\< 84 days prior to registration * ARM B: Uncontrolled infection * ARM B: Systemic anti-cancer therapy =\< 4 weeks prior to registration * ARM B: Known HIV infection * ARM B: Other concurrent chemotherapy, immunotherapy, radiotherapy, or investigational therapy * ARM B: Pregnant or nursing women * ARM B: History of bone marrow or solid organ transplantation * ARM B: Patient for whom surgical resection or liver transplantation would be more appropriate * ARM B: Any condition, which in the opinion of the investigator would render the patient unsuitable to participate in the study * ARM B: Any corticosteroid use =\< 28 days prior to registration * ARM B: Any radioembolization or TACE =\< 84 days prior to registration