A massive increase in the amount of data traffic over mobile wireless communication has been observed in recent years, while further rapid growth is expected in the years ahead. The current fourth-generation (4G) mobile networks are evolving to the fifth-generation (5G) networks to fulfill the demand for high data rates and broad network coverage. The advent of the ultra-dense 5G network and a vast number of connected devices will bring about the obvious issues of significantly increased system energy consumption, operational expenses, and carbon dioxide emissions. Renewable energy is considered a viable and practical approach to power the small cell base station in an ultra-dense 5G network infrastructure to reduce the energy provisions from the electric grid and carbon dioxide emissions. In this paper, we discuss the role of renewable energy in the design of sustainable, eco-friendly, and cost-effective 5G mobile networks and provide a comprehensive survey on the state-of-art of renewable energy management techniques aiming to promote the sustainability and cost reduction of the large-scale mobile wireless infrastructures. This survey specifically covers a variety of energy efficiency techniques, the utilization of renewable energy sources, interaction with the smart grid (SG), and the renewable energy powered base stations. It also highlights the outstanding technical challenges and future perspectives to enable future sustainable 5G network infrastructure.