The Internet of things (IoT) is an active area in the current research community due to the improvement in mobile computing and wireless networks. Currently, the IoT is involved in many fields like smart cities, smart health monitoring, smart tracking, and smart factory; therefore, it is introducing new research opportunities and industrial revolutions. Smart health, in particular, is very important and trendy domain for researchers and practitioners due to its continuous monitoring of health of patients. The objective of smart health is to provide medical facilities to patients at anytime and anywhere. The smart health monitoring systems are mostly connected with the wireless network medium that is extremely vulnerable for threats. However various attacks are observed that can endanger these health monitoring applications and systems. These attacks include Denial of Service (DoS) Attack, Fingerprint and Timing-based Snooping, Router Attack, Select and Forwarding attack, Sensor attack and Replay Attack. In this paper, we discuss these attacks with their impact on health monitoring systems with some suggestive measures from our research findings.