Hardware impairments (HI) are always present in low-cost wireless devices. This paper investigates the outage behaviors of intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) networks by taking into account the impact of HI. Specifically, we derive the approximate and asymptotic expressions of the outage probability for the IRS-NOMA-HI networks. Based on the asymptotic results, the diversity orders under perfect self-interference cancellation and imperfect self-interference cancellation scenarios are obtained to evaluate the performance of the considered network. In addition, the system throughput of IRS-NOMA-HI is discussed in delay-limited mode. The obtained results are provided to verify the accuracy of the theoretical analyses and reveal that: 1) The outage performance and system throughput for IRS-NOMA-HI outperforms that of the IRS-assisted orthogonal multiple access-HI (IRS-OMA-HI) networks; 2) The number of IRS elements, the pass loss factors, the Rician factors, and the value of HI are pivotal to enhancing the performance of IRS-NOMA-HI networks; and 3) It is recommended that effective methods of reducing HI should be used to ensure system performance, in addition to self-interference cancellation techniques.