In this article, we explore the application of hybrid automatic repeat request (HARQ) within the intelligent reflection surface-assisted nonorthogonal multiple access (IRS-NOMA) system. We investigate the closed-form expressions for the outage probability of multiple users in the HARQ-assisted IRS-NOMA system, considering scenarios with perfect successive interference cancellation (pSIC) and imperfect successive interference cancellation (ipSIC), respectively. A definite integral approximation method for multidimensional functions based on Gauss-Chebyshev quadrature (GCQ) is proposed to conduct the performance analysis of the proposed HARQ-assisted IRS-NOMA system. Based on the asymptotic outage probability, the diversity order of multiple users for HARQ-assisted IRS-NOMA is obtained. Under the analytical results, the diversity order of the mth $(m\gt 1)$ user for the HARQ-assisted IRS-NOMA with ipSIC is zero, and that of the mth user for the HARQ-assisted IRS-NOMA with pSIC is in connection with the number of reflecting elements and the number of transmission rounds. The simulation results are presented to substantiate the accuracy of the analytical results. The results demonstrate that: 1) the HARQ-assisted IRS-NOMA systems can achieve a significant gain compared with the IRS-NOMA systems; 2) the HARQ-assisted IRS-NOMA outperforms the HARQ-assisted IRS-orthogonal multiple access in terms of outage probability and diversity order; and 3) the HARQ with incremental redundancy (HARQ-IR)-assisted IRS-NOMA system has a better performance than the HARQ with chase combining (HARQ-CC)-assisted IRS-NOMA system.