The healthcare industry is rapidly adapting to new computing environments and technologies. With academics increasingly committed to developing and enhancing healthcare solutions that combine the Internet of Things (IoT) and edge computing, there is a greater need than ever to adequately monitor the data being acquired, shared, processed, and stored. The growth of cloud, IoT, and edge computing models presents severe data privacy concerns, especially in the healthcare sector. However, rigorous research to develop appropriate data privacy solutions in the healthcare sector is still lacking. This paper discusses the current state of privacy-preservation solutions in IoT and edge healthcare applications. It identifies the common strategies often used to include privacy by the intelligent edges and technologies in healthcare systems. Furthermore, the study addresses the technical complexity, efficacy, and sustainability limits of these methods. The study also highlights the privacy issues and current research directions that have driven the IoT and edge healthcare solutions, with which more insightful future applications are encouraged.