The growing demanding data rate requirements of future wireless networks have raised enormous concerns on energy consumption. Researchers conduct their study of green communication in terms of wireless communication environment and low power consumption communication technique. Recently, intelligent reflecting surface (IRS) and ambient backscatter communication (ABC) have been proposed, respectively, and they have attracted growing attention. In this paper, we study the ABC-enabled non-orthogonal multiple access (NOMA) networks. In such networks, environment obstacles may block the communication from the base station to the cell-edge user. To overcome this drawback, we propose an IRS-assisted relay transmission scheme. Therefore, in the ABC-enabled IRS-NOMA networks, there is a problem of how to enhance the network performance by optimizing the phase-shifts as well as the power resource allocation. We formulate an optimization problem to maximize the energy efficiency by jointly optimizing the phase-shifts of IRS’s reflecting elements and the power allocation of NOMA users. To deal with this problem, it is divided into two subproblems: 1) phase-shifts optimization and 2) power allocation optimization. The Dinkelbach-based iterative algorithm is applied to obtain maximum energy efficiency. Simulation results show that our proposed scheme always performs the best in terms of energy efficiency and throughput.