Recent developments in sensors, wireless communications, and data processing technologies are the main drivers for adopting the Internet of Things (IoT) in healthcare systems. IoT-based healthcare systems can enhance the quality of life significantly and help prevent the occurrence of health problems and epidemics. Deploying IoT-based healthcare on a massive scale raises several issues and challenges. One of the main challenges is the management of the end-to-end network connections of the IoT-based healthcare system. This paper presents a comprehensive survey of smart network management protocols that improve IoT-based healthcare efficiency, ensuring real-time monitoring, secure data transmission, and effective device management. Moreover, a reference architecture has been proposed for the network management of IoT-based smart healthcare systems to ensure the sustainability of service delivery to patients and caregivers. The architecture avoids health-related risks and anomalies by incorporating proper network management techniques and operational requirements pertaining to smart healthcare systems. This paper also discusses architectural implementation insights supported by new technologies such as software-defined networking (SDN) and deep learning (DL). Finally, this paper explores emerging paradigms to advance next-generation network management protocols for future smart healthcare systems.