Ultra-dense Internet of Things (IoT) network has greatly facilitated the development of smart environments and the realization of diverse sophisticated applications. Power constrained and resource limited IoT devices often need to perform computation-intensive and delay-sensitive tasks in such a network. Mobile edge computing and non-orthogonal multiple access are two promising techniques to address the corresponding challenges. In this paper, in order to improve the fairness and resource efficiency among IoT users, resource allocation problems are formulated in ultra-dense MEC-enabled IoT networks with NOMA considered. An iterative algorithm based on successive convex approximation technique is proposed to solve those challenging non-convex problems. Simulation results show that our proposed allocation method outperforms the benchmark schemes and verifies the efficiency of the proposed algorithm.