In this study, we revisit the commonly-cited off-target issue in multilingual neural machine translation (MNMT). By carefully designing experiments on different MNMT scenarios and models, we attribute the off-target issue to the overfitting of the shortcuts of (non-centric, centric) language mappings. Specifically, the learned shortcuts biases MNMT to mistakenly translate non-centric languages into the centric language instead of the expected non-centric language for zero-shot translation. Analyses on learning dynamics show that the shortcut learning generally occurs in the later stage of model training, and multilingual pretraining accelerates and aggravates the shortcut learning. Based on these observations, we propose a simple and effective training strategy to eliminate the shortcuts in MNMT models by leveraging the forgetting nature of model training. The only difference from the standard training is that we remove the training instances that may induce the shortcut learning in the later stage of model training. Without introducing any additional data and computational costs, our approach can consistently and significantly improve the zero-shot translation performance by alleviating the shortcut learning for different MNMT models and benchmarks.