Deploying models across diverse devices demands tradeoffs among multiple objectives due to different resource constraints. Arguably, due to the small model trap problem in multi-objective neural architecture search (MO-NAS) based on a supernet, existing approaches may fail to maintain large models. Moreover, multi-tasking neural architecture search (MT-NAS) excels in handling multiple tasks simultaneously, but most existing efforts focus on tasks from the same dataset, limiting their practicality in real-world scenarios where multiple tasks may come from distinct datasets. To tackle the above challenges, we propose a Multi-Objective Evolutionary Multi-Tasking framework for NAS (MO-EMT-NAS) to achieve architectural knowledge transfer across tasks from different datasets while finding Pareto optimal architectures for multi-objectives, model accuracy and computational efficiency. To alleviate the small model trap issue, we introduce an auxiliary objective that helps maintain multiple larger models of similar accuracy. Moreover, the computational efficiency is further enhanced by parallelizing the training and validation of the weight-sharing-based supernet. Experimental results on seven datasets with two, three, and four task combinations show that MO-EMT-NAS achieves a better minimum classification error while being able to offer flexible trade-offs between model performance and complexity, compared to the state-of-the-art single-objective MT-NAS algorithms. In addition, the runtime of MO-EMT-NAS is reduced by 59.7% to 77.7%, compared to the corresponding multi-objective single-task approaches.