Cixutumumab and Sorafenib Tosylate in Treating Patients With Advanced Liver Cancer

Inclusion Criteria: * Unresectable or metastatic HCC for which standard curative measures do not exist; the diagnosis of hepatocellular carcinoma should be based on at least one of the following: * The presence of one or more liver lesions, measuring ≥ 2 cm, with characteristic arterial enhancement and venous washout in the setting of liver cirrhosis and/or hepatitis B or C infection * The presence of liver lesion(s) with AFP \>= 400 * Tissue confirmation in the absence of a and/or b * Tissue availability is desired and will be sought, but tissue availability is not mandated for accrual to the study * No prior systemic therapy for HCC; patients may have had prior embolization, chemoembolization, intra-arterial chemotherapy infusion, ethanol injection, radiofrequency ablation or cryosurgery * ECOG 0 or 1 * Life expectancy of greater than 3 months * Absolute neutrophil count \> 1,000/mm\^3 * Platelets \> 65,000/mm\^3 * Total bilirubin =\< 2 x the institutional upper normal limit * AST and ALT =\< 5 x the institutional upper normal limit * Renal function =\< 1.5 mg/dl or calculated creatinine clearance \> 50 mL/min (Cockcroft-Gault formula) * PT \< 4 seconds of prolongation above the upper normal limit * No evidence of encephalopathy in the last 6 months * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; 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 * Ability to understand and willing to sign a written informed consent document Exclusion Criteria: * Local therapy for HCC within 4 weeks prior to treatment on this study or those who have not recovered from adverse events related to therapy administered more than 4 weeks earlier * Receiving other investigational agents * Brain metastases, because of their poor prognosis, proclivity for progressive neurologic dysfunction that would confound the evaluation of neurologic adverse events, and the potential for increased risk for CNS adverse events * Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated on this clinical trial * HIV-positive patients are ineligible * Fasting blood glucose \> 160 mg/dL * Esophageal or gastric variceal bleeding within the last 6 * Clinically evident ascites (minimal, medically controlled ascites detectable on imaging studies only is allowed) * Child-Pugh C cirrhosis or Child-Pugh B cirrhosis with more than 7 points * Patients unable to swallow the sorafenib tablets whole are ineligible; (the tablets cannot be crushed or broken) * Cardiac: symptomatic congestive heart failure, unstable angina, clinically significant and uncontrolled cardiac dysrhythmia, uncontrolled hypertension (systolic BP \> 150 or diastolic BP \> 100 on two occasions within two weeks of beginning therapy on this protocol, myocardial infarction within 6 months, NYHA class \> II, LVEF \< normal as assessed on MUGA * Fibrolamellar carcinoma or any mixed variants of HCC with fibrolamellar histology * Hypersensitivity to human IgG unless the patient has subsequently tolerated IgG agents * Patients with active hepatitis B infection should be on adequate antiviral therapy