Telehealth Extension Act of 2021 This bill expands and otherwise modifies coverage of telehealth services under Medicare. Specifically, the bill permanently (1) removes geographic restrictions on originating sites (i.e., the location of the beneficiary), (2) allows the home of the beneficiary to serve as the originating site for all services, and (3) allows federally qualified health centers and rural health clinics to serve as the distant site (i.e., the location of the health care practitioner). The bill also extends any Medicare telehealth flexibilities that were granted during the COVID-19 public health emergency until two years after the emergency ends and generally allows such flexibilities to be granted during any public health emergency. In addition, the bill (1) allows for Medicare payment of outpatient critical access hospital services consisting of telehealth behavioral therapy until two years after the COVID-19 public health emergency ends; and (2) after the emergency ends, conditions payment for certain high-cost laboratory tests and durable medical equipment that are ordered via telehealth on at least one in-person visit during the preceding six-month period.