3D object recognition has successfully become an appealing research topic in the real-world. However, most existing recognition models unreasonably assume that the categories of 3D objects cannot change over time in the real-world. This unrealistic assumption may result in significant performance degradation for them to learn new classes of 3D objects consecutively, due to the catastrophic forgetting on old learned classes. Moreover, they cannot explore which 3D geometric characteristics are essential to alleviate the catastrophic forgetting on old classes of 3D objects. To tackle the above challenges, we develop a novel Incremental 3D Object Recognition Network (i.e., InOR-Net), which could recognize new classes of 3D objects continuously via overcoming the catastrophic forgetting on old classes. Specifically, a category-guided geometric reasoning is proposed to reason local geometric structures with distinctive 3D characteristics of each class by leveraging intrinsic category information. We then propose a novel critic-induced geometric attention mechanism to distinguish which 3D geometric characteristics within each class are beneficial to overcome the catastrophic forgetting on old classes of 3D objects, while preventing the negative influence of useless 3D characteristics. In addition, a dual adaptive fairness compensations strategy is designed to overcome the forgetting brought by class imbalance, by compensating biased weights and predictions of the classifier. Comparison experiments verify the state-of-the-art performance of the proposed InOR-Net model on several public point cloud datasets.