A novel framework of the unmanned aerial vehicle (UAV)-mounted active simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) communications with the non-orthogonal multiple access (NOMA) is proposed for Internet-of-Things (IoT) networks. In particular, an active STAR-RIS is deployed onboard to enhance the communication link between the base station (BS) and the IoT devices, and NOMA is utilized for supporting the multi-device connectivity. Based on the proposed framework, a system sum rate maximization problem is formulated for the joint optimization of the active STAR-RIS beamforming, the UAV trajectory design, and the power allocation. To solve the non-convex problem with highly-coupled variables, an alternating optimization (AO) algorithm is proposed to decouple the original problem into three subproblems. Specifically, for the active STAR-RIS beamforming, the amplification coefficient, the power-splitting ratio, and the phase shift are incorporated into a combined variable to simplify the optimization process. Afterwards, the penalty-based method is invoked for handling the non-convex rank-one constraint. For the UAV trajectory design and the power allocation subproblems, the successive convex optimization method is applied for iteratively approximating the local-optimal solution. Numerical results demonstrate that: 1) the proposed algorithm achieves superior performance compared to the benchmarks in terms of the sum rate; and 2) the UAV-mounted active STAR-RIS can effectively enhance the channel gain from the BS to the IoT devices by the high-quality channel construction and the power compensation.