Toxic Exposure Safety Act of 2021 This bill expands an existing program that compensates Department of Energy (DOE) employees and contractors for illnesses caused by occupational exposure to radiation and certain substances. Specifically, the bill requires the Energy Employees Occupational Illness Compensation Program to cover additional workers with illnesses caused by exposure to toxic substances at certain DOE facilities, including cleanup sites. The Department of Health and Human Services must establish rules for determining whether a current or former DOE employee or contractor is entitled to compensation. In addition, the Department of Labor must create or update assessments of toxic substance exposure in each DOE facility, as well as carry out outreach activities to provide DOE workers with information about the program. The bill also appropriates funds as necessary for DOE's assistance with certain Labor activities related to DOE facilities. Furthermore, the bill requires research on diseases associated with toxic substance exposure at DOE facilities.