Increasing Rural Telehealth Access Act of 2021 This bill establishes, as part of certain grants for telehealth networks and resource centers, a pilot project to increase the use of remote patient monitoring technology in rural areas. Specifically, the Health Resources and Services Administration (HRSA) must award grants to coordinate care in rural areas for individuals with chronic conditions and conduct other activities using remote patient monitoring technology. To be eligible for a grant, entities must meet requirements for participating in telehealth networks and must use appropriate technologies. In awarding these grants, HRSA shall give priority to entities that are capable of establishing programs quickly and that use technologies that provide continuous, real-time coaching services.