Tactile Internet is an emerging concept that focuses on supporting high-fidelity, ultra-responsive, and widely available human-to-machine interactions. To reduce the transmission latency and alleviate Internet congestion, fog computing has been advocated as an important component of the Tactile Internet. In this paper, we focus on energy-efficient design of fog computing networks that support low-latency Tactile Internet applications. We investigate two performance metrics: Service response time of end-users and power usage efficiency of fog nodes. We quantify the fundamental tradeoff between these two metrics and then extend our analysis to fog computing networks involving cooperation between fog nodes. We introduce a novel cooperative fog computing concept, referred to as offload forwarding, in which a set of fog nodes with different computing and energy resources can cooperate with each other. The objective of this cooperation is to balance the workload processed by different fog nodes, further reduce the service response time, and improve the efficiency of power usage. We develop a distributed optimization framework based on dual decomposition to achieve the optimal tradeoff. Our framework does not require fog nodes to disclose their private information nor conduct back-and-forth negotiations with each other. Two distributed optimization algorithms are proposed. One is based on the subgradient method with dual decomposition and the other is based on distributed ADMM-VS. We prove that both algorithms can achieve the optimal workload allocation that minimizes the response time under the given power efficiency constraints of fog nodes.