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Unmanned aerial vehicle (UAV)-enabled edge computing is emerging as a potential enabler for Artificial Intelligence of Things (AIoT) in the forthcoming sixth-generation (6G) communication networks. With the use of flexible UAVs, massive sensing data is gathered and processed promptly without considering geographical locations. Deep neural networks (DNNs) are becoming a driving force to extract valuable information from sensing data. However, the lightweight servers installed on UAVs are not able to meet the extremely high requirements of inference tasks due to the limited battery capacities of UAVs. In this work, we investigate a DNN model placement problem for AIoT applications, where the trained DNN models are selected and placed on UAVs to execute inference tasks locally. It is impractical to obtain future DNN model request profiles and system operation states in UAV-enabled edge computing. The Lyapunov optimization technique is leveraged for the proposed DNN model placement problem. Based on the observed system overview, an advanced online placement (AOP) algorithm is developed to solve the transformed problem in each time slot, which can reduce DNN model transmission delay and disk I/O energy cost simultaneously while keeping the input data queues stable. Finally, extensive simulations are provided to depict the effectiveness of the AOP algorithm. The numerical results demonstrate that the AOP algorithm can reduce 18.14% of the model placement cost and 29.89% of the input data queue backlog on average by comparing it with benchmark algorithms.