Preventing Health Emergencies And Temperature-related Illness and Deaths Act of 2021 or the Preventing HEAT Illness and Deaths Act of 2021 This bill provides statutory authority for, and otherwise revises, the National Integrated Heat Health Information System to improve the capacity of the United States to prepare for, adapt to, and mitigate health risks of extreme heat. The bill tasks the National Oceanic and Atmospheric Administration (NOAA) with administering the system. (Currently, the system is jointly administered by NOAA and the Centers for Disease Control and Prevention.) The bill also authorizes financial assistance for projects to increase awareness, education, and research about extreme heat events and prioritizes projects that serve historically disadvantaged communities and communities with the greatest risk or highest incidence of heat-related illnesses and deaths. Additionally, NOAA must contract with the National Academies of Sciences, Engineering, and Medicine to study relevant policy, research, and data gaps concerning extreme heat information and response. The bill also establishes an interagency committee within the White House Office of Science and Technology Policy to develop a strategic plan and otherwise coordinate federal activities related to addressing health risks and impacts of heat.