The last few decades have witnessed an explosive growth of the Internet-of-Things (IoT) systems, which provide ubiquitous sensing and computing services. When adopted in industrial and manufacturing environments, IoT is referred to as the industrial IoT (IIoT), which has attracted increasing research attention. Energy efficiency is one of the most important research topics in green IIoT, as 1) the limited resource can significantly affect the lifetime of IIoT systems and 2) massive sensors, devices, machines keep consuming a considerable amount of energy, and increasing the carbon footprint. In this article, we present a comprehensive survey on energy-efficient communications and computation mechanisms in IIoT systems (such as smart grids). We categorize the existing works, review, discuss, and compare the works to explore their pros and cons. We also discuss the open issues and research challenges, considering the recent 5G communications and edge computing trends.