The rapid development of Internet of Things (IoT)-enabled systems in public and private spaces offers consumers numerous conveniences. Among different Internet-connected systems, the use of e-health systems is growing rapidly. The utilization of IoT devices and cloud-fog network technologies has made e-healthcare provision more convenient. While providing valuable services to the healthcare sector, like any other IoT-enabled systems it is putting pressure on energy, an essential element of life. Therefore, it is imperative to know the energy consumption model of e-health systems. Considering the importance of energy consumption in IoT-based systems, this article develops a cloud-fog-based e-health system and makes it energy efficient by understanding energy consumption at different layers of communication. Moreover, how fog integration with the cloud reduces energy consumption and delays at different stages of communication is discussed.