In this paper, we study the resource allocation problem for downlink multicarrier non-orthogonal multiple access systems with short-packet communications (MC-NOMA-SPC). In contrast to long-packet communications in conventional wireless systems, SPC suffers from a transmission rate degradation and an unavoidable decoding error rate. Thus conventional resource allocation based on the Shannon capacity assuming infinite blocklength is no longer optimal. In this paper, we propose to use the effective-throughput as the performance metric to evaluate the tradeoff between the transmission rate and the decoding error rate. Building on that, we jointly optimize the subcarrier assignment, transmission power allocation, and transmission rate adaptation of each user to maximize the total weighted effective-throughput subject to transmission reliability and various practical constraints. Since the formulated problem belongs to a non-convex mixed integer nonlinear programming (MINLP) problem, we develop an efficient algorithm based on the dynamic programming (DP) recursion framework to obtain its optimal solutions. We further propose a low-complexity algorithm based on the principles of block coordinate descent (BCD) and concave-convex procedure (CCCP). Finally, the simulation results show that the proposed low-complexity algorithm can achieve similar performance as that of the optimal solution and outperform the other baseline schemes significantly.