As a flexible and scalable architecture, heterogeneous cloud radio access networks (H-CRANs) inject strong vigor into the green evolution of current wireless networks. But the brutal truth is that energy efficiency (EE) improves at the cost of other indexes such as spectral efficiency (SE), fairness, and delay. It is thus important to investigate performance tradeoffs for striking flexible balances between energy-efficient transmission and excellent quality-of-service (QoS) guarantees under this new architecture. In this article, we first propose some potential techniques to energy-efficiently operate H-CRANs by exploiting their features. We then elaborate the initial ideas of modeling three fundamental tradeoffs, namely EE-SE, EE-fairness, and EE-delay tradeoffs, when applying these green techniques, and present open issues and challenges for future investigations. These related results are expected to shed light on green operation of H-CRANs from adaptive resource allocation, intelligent network control, and scalable network planning.