Background: Sustainability in healthcare has gained increasing importance due to its impact on environmental, financial, and social strategies, as well as on public health, and therefore, relevant policies and actions can also play a significant role in telemedicine services. The establishment of a sustainable telemedicine network at a country level is important to improve access to healthcare, reduce costs, increase convenience, and ensure the continuity of service delivery. However, there are significant environmental, social, technological, human, and governance challenges to meet the sustainability conditions for these networks. Methods: Thus, a narrative literature review was conducted to investigate the telemedicine implementation aspects and the sustainability dimensions in a unified approach and integrated strategy in order to develop a more resilient and equitable healthcare solution, ensuring its long-term integration into healthcare systems. Results: This paper aims to identify critical factors related to the proposed governance model for the National Telemedicine Network in Greece (EDIT) that influence sustainability requirements and interdisciplinary strategies to address relevant challenges. Conclusions: By examining these factors, the paper seeks to propose the fundamental pillars of a sustainable telemedicine framework and the methodology for developing a sustainability plan that will enhance EDIT’s capacities toward a sustainable and resilient operation of telemedicine as a standard practice within the Greek healthcare system.