Rural Hospital Closure Relief Act of 2023 This bill allows additional hospitals to qualify as critical access hospitals (CAHs) that receive special payment under Medicare. Currently, in order to qualify as a CAH under Medicare, a hospital must either (1) be located more than 35 miles (or 15 miles in mountainous regions or areas with only secondary roads) from another hospital, or (2) have been certified prior to January 1, 2006, by the state as a necessary provider of services in the area. The bill allows a hospital to also qualify if the hospital is a small, rural hospital that (1) serves a health professional shortage area, or a high number of low-income individuals or Medicare or Medicaid beneficiaries; (2) has experienced financial losses for two consecutive years; and (3) attests to having a strategic plan to address financial solvency. The Government Accountability Office must study the effects of the bill's implementation. The Centers for Medicare & Medicaid Services must subsequently establish a mechanism and issue guidance on how newly designated CAHs may transition to different payment models under Medicare.