Fly Safe and Healthy Act of 2021 This bill directs the Transportation Security Administration (TSA) to establish a 120-day pilot program to conduct temperature checks for domestic and international passengers, individuals accompanying those passengers, crew members, and other individuals who pass through airports and airport security screening locations. In developing the pilot program, the TSA must address certain policies and procedures, including (1) accommodating individuals with disabilities or observing certain religious practices, and (2) exempting individuals who may have a fever unrelated to COVID-19 (i.e. coronavirus disease 2019). The pilot program must ensure airlines allow passengers who are prohibited from flying due to a fever or a secondary medical screening to reschedule or cancel a flight at no cost. The TSA must, within 90 days after the completion of the pilot program, create a policy for deploying a temperature check program at airports and airport security screening locations through the end of the COVID-19 public health emergency. The Department of Transportation must revise its regulations to require the inclusion of certain information about traveling in the passenger notification system during the COVID-19 public health emergency, including (1) a message discouraging any individual who has a fever from traveling in air transportation, and (2) a notification that each passenger and any individual accompanying a passenger into the sterile area of the airport will undergo a temperature check if the pilot program established under this bill is in effect in that airport.