Toxic Exposure Safety Act of 2020 This bill expands an existing program to provide compensation to Department of Energy (DOE) employees and contractors for illnesses caused by occupational exposure to radiation and certain substances to cover additional workers with illnesses caused by toxic substances. The Energy Employees Occupational Illness Compensation Program shall cover DOE employees and contractors who are members of the Toxic Special Exposure Cohort established under this bill. The Department of Health and Human Services shall establish rules for determining whether a DOE employee or contractor qualifies as a cohort member, including a list of qualifying diseases. The bill also requires the Department of Labor to create or update assessments of toxic substance exposure in each DOE facility. The bill also appropriates funds as necessary for DOE's assistance with certain Labor activities related to DOE facilities. Furthermore, the National Institute of Environmental Health Sciences shall conduct or support research on the epidemiological impacts of toxic substance exposure at DOE facilities.