This paper studies the performance of intelligent reflecting surface (IRS)-aided short-packet non-orthogonal multiple access (NOMA) systems under perfect and imperfect successive interference cancellation. Closed-form expressions for the average block error rate (BLER) with random and optimal phase shifts are derived. To gain some useful insights, an asymptotic analysis is carried out, based on which the diversity gain is achieved. Numerical results show that, at high signal-to-noise ratio, users with random phase shift achieve significant performance improvements in the performance gap between direct and non-direct links from a source when compared with that of the optimal phase shift. In addition, the proposed system outperforms its orthogonal multiple access counterpart in terms of both BLER and throughput. Furthermore, optimal channel coding rates of users can maximize the system throughput performance. Monte Carlo simulations are presented to corroborate the theoretical analysis.