Advancements in the Internet of Medical Things (IoMT) have made remote patient monitoring increasingly viable. However, challenges persist in safeguarding sensitive data, optimizing resources, and addressing the energy constraints of patient devices. This paper presents a health monitoring framework integrating Mobile Edge Computing (MEC) and sixth-generation (6G) technologies, structured into internal Medical Body Area Networks (int-MBANs) and external communications beyond MBANs (ext-MBANs). For int-MBANs, the proposed OptiBand algorithm optimizes energy consumption, extends device standby time, and considers message timeliness and medical criticality. A key innovation of OptiBand is its incorporation of patient’s device standby time into the resource allocation strategy to address real-world patient needs. For ext-MBANs, the DynaMEC algorithm dynamically balances energy efficiency, privacy protection, latency, and fairness, even under varying patient scales. A latency-aware scheduling mechanism also be introduced to guarantee timely completion of emergency tasks. Theoretical analysis and experimental results confirm the feasibility, convergence, and optimality of both algorithms. These characteristics and advantages of the proposed system make remote patient monitoring through IoMT more feasible and effective.