Network Function Virtualization (NFV) provides an effective way to reduce the network provider's cost by allowing multiple Virtual Networks (VNs) to share the underlying physical infrastructure. In the NFV environment, especially when supporting multicast service over the VNs, reliability is a critical requirement in the process of VN mapping since the failure of one virtual node can cause the malfunction of all the subsequent nodes that receive multicasting data from it. In this paper, for the first time, we study how to efficiently map VNs for reliable multicast services, while taking into consideration the max-min fairness of the reliability among distinct VNs. We propose a Mixed Integer Linear Programming (MILP) model to determine the upper bound on the max-min fairness reliability. In addition, an efficient heuristic, namely Uniform Reliability Mutation based Genetic (URMG) algorithm, is developed to address reliable multicast VN mapping with a low computational complexity. By encoding multicast tree construction and link mapping into path selection, taking into consideration the max-min reliability fairness goal, and the networking reliability factors during mutation, URMG can globally optimize the reliability and its fairness of all the multicast VN requests. Through extensive simulations, we demonstrate that URMG achieves close to the optimal reliability fairness with a much lower time complexity than the MILP and yields a significant performance improvement in terms of reliability fairness, bandwidth consumption and transmission delay comparing with other heuristic solutions.