Device-to-device (D2D) and non-orthogonal multiple access (NOMA) are promising technologies to meet the challenges of the next generations of mobile communications in terms of network density and diversity for internet of things (IoT) services. This paper tackles the problem of maximizing the D2D sum-throughput in an IoT system underlaying a cellular network, through optimal channel and power allocation. NOMA is used to manage the interference between cellular users and full-duplex (FD) IoT devices. To this aim, mutual successive interference cancellation (SIC) conditions are identified to allow simultaneously the removal of the D2D devices interference at the level of the base station and the removal of the cellular users (CU) interference at the level of D2D devices. To optimally solve the joint channel and power allocation (PA) problem, a time-efficient solution of the PA problem in the FD context is elaborated. By means of graphical representation, the complex non-convex PA problem is efficiently solved in constant time complexity. This enables the global optimal resolution by successively solving the separate PA and channel assignment problems. The performance of the proposed strategy is compared against the classical state-of-the-art FD and HD scenarios, where SIC is not applied between CUs and IoT devices. The results show that important gains can be achieved by applying mutual SIC NOMA in the IoT-cellular context, in either HD or FD scenarios.