Fly Safe and Healthy Act of 2021 This bill directs the Transportation Security Administration (TSA) to establish a 180-day pilot program to conduct temperature checks for domestic and international passengers, individuals accompanying those passengers, crew members, and other individuals who pass through airport security screening locations at which screening is conducted by the TSA. In developing the pilot program, the TSA must address certain policies and procedures, including (1) types and costs of technology that may be used to conduct temperature checks, (2) mechanisms to protect the privacy and medical information of individuals subjected to temperature checks, and (3) exempting individuals who may have a fever unrelated to COVID-19. The pilot program must ensure airlines allow passengers who are prohibited from flying due to a fever or a secondary medical screening to reschedule, request a voucher, or cancel a flight at no cost. The TSA must design, test, and implement a passenger notification system for alerting passengers of temperature checks under the pilot program that includes (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, subject to a secondary medical screening, if the pilot program established under this bill is in effect in that airport.