The rapid growth of the elderly population in the world and the rising cost of healthcare impose big issues for healthcare and medical monitoring. A Wireless Body Sensor Network (WBSN) is comprised of small sensor nodes attached inside, on or around a human body, the main purpose of which is to monitor the functions and surroundings of the human body. However, the heat generated by the node's circuitry and antenna could cause damage to the human tissue. Therefore, in designing a routing protocol for WBSNs, it is important to reduce the heat by incorporating temperature into the routing metric. The main contribution of this paper is to survey existing temperature-aware routing protocols that have been proposed for WBSNs. In this paper, we present a brief overview of WBSNs, review the existing routing protocols comparatively and discuss challenging open issues in the design of routing protocols.