Lassa virus is estimated to cause thousands of human deaths per year, primarily due to spillovers from its natural host, Mastomys rodents. Efforts to create vaccines and antibody therapeutics must account for the evolutionary variability of Lassa virus's glycoprotein complex (GPC), which mediates viral entry into cells and is the target of neutralizing antibodies. To map the evolutionary space accessible to GPC, we use pseudovirus deep mutational scanning to measure how nearly all GPC amino-acid mutations affect cell entry and antibody neutralization. Our experiments define functional constraints throughout GPC. We quantify how GPC mutations affect neutralization by a panel of monoclonal antibodies and show that all antibodies are escaped by mutations that exist among natural Lassa virus lineages. Overall, our work describes a biosafety-level-2 method to elucidate the mutational space accessible to GPC and shows how prospective characterization of antigenic variation could aid design of therapeutics and vaccines.