With the global outbreak of COVID-19, the Internet of Medical Things (IoMT), as an extension of the Internet of Things (IoT), has received increasing attention due to its ability to remotely monitor the main symptoms. In addition, based on widely collected IoMT data, the spread and even the origin of a pandemic can be traced effectively. However, spectrum resources become scarce with the increasing number of patients. Therefore, the effective allocation of communication, computing, and cache (3C) resources becomes increasingly important. In this article, we first study the unique characters of IoMT and construct a distributed 3C resource framework. Interactions of gateways in IoMT are specified and formed as an infinitely repeating game, where each gateway performs strategy selection based on probability learning at the network edge to reach a Nash equilibrium. A case study is provided to demonstrate the high efficiency of the designed 3C resource allocation strategy. Finally, we discuss the challenges and open issues of edge learning with IoMT from three aspects: real-time edge learning, 3C resource management, and privacy.