In this paper, we examine the robust beamforming design to tackle the energy efficiency (EE) maximization problem in a 5G massive multiple-input multiple-output (MIMO)-non-orthogonal multiple access (NOMA) downlink system with imperfect channel state information (CSI) at the base station. A novel joint user pairing and dynamic power allocation (JUPDPA) algorithm is proposed to minimize the inter user interference and also to enhance the fairness between the users. This work assumes imperfect CSI by adding uncertainties to channel matrices with worst-case model, i.e., ellipsoidal uncertainty model (EUM). A fractional non-convex optimization problem is formulated to maximize the EE subject to the transmit power constraints and the minimum rate requirement for the cell edge user. The designed problem is difficult to solve due to its nonlinear fractional objective function. We firstly employ the properties of fractional programming to transform the non-convex problem into its equivalent parametric form. Then, an efficient iterative algorithm is proposed established on the constrained concave-convex procedure (CCCP) that solves and achieves convergence to a stationary point of the above problem. Finally, Dinkelbach’s algorithm is employed to determine the maximum energy efficiency. Comprehensive numerical results illustrate that the proposed scheme attains higher worst-case energy efficiency as compared with the existing NOMA schemes and the conventional orthogonal multiple access (OMA) scheme.