Telemedicine Monitoring (TM) integrates mobile communication technology and Internet of Things (IoT) technology for health monitoring and data management. Amidst the escalating demand for telemedicine, traditional cloud computing struggles to guarantee real-time performance and data privacy. To address these challenges, we systematically survey the application of fog and edge computing technologies in TM systems. We focus on the following key aspects: (1) We delve into the theoretical foundations of fog and edge computing, underscoring their salient advantages including low latency, location awareness, high mobility, and more. (2) We elaborate on the architecture of a TM system hinged on fog and edge computing. (3) We outline key challenges facing fog/edge computing-based TM systems, including bandwidth limitations, low latency, data security, privacy, heterogeneity, and reliability. (4) We discuss the need for future advancements in the realms of security defense capability, system adaptability, and convergence of scheduling algorithms to refine the construction of the TM system and stimulate the development of telemedicine.