Most existing 3D object classification and retrieval algorithms rely on one-off supervised learning on closed 3D object sets and tend to provide rigid convolutional neural networks with little scalability. Such limitations substantially restrict their potential to learn newly emerged 3D object classes continually in the real world. Aiming to go beyond these limitations, we innovatively propose two new and challenging tasks: class-incremental 3D object classification (CI-3DOC) and class-incremental 3D object retrieval (CI-3DOR), the key to which is class-incremental 3D representation learning. It expects the network to update continually to learn new 3D class representations without forgetting the previously learned ones. To this end, we design a novel balanced distillation network(BDNet)that uses a dual supervision mechanism to balance between consolidating old knowledge (stability) and adapting to new 3D object classes (plasticity) carefully. On the one hand, we employ stability-based supervision to retain the stable and discriminative information of old classes that greatly benefit both classification and retrieval tasks. On the other hand, we use plasticity-based supervision to improve the network's generalization for learning new class 3D representations by transferring knowledge from a temporary teacher network to the current model. By properly handling the relationship between the two modules, we achieve a surprising performance improvement. Furthermore, considering there is no available dataset for evaluation, we build two 3D datasets, INOR-1 and INOR-2, to evaluate these two new tasks. Extensive experimental results demonstrate that our method can significantly outperform other state-of-the-art class-incremental learning methods. Even if we store 500-1000 fewer 3D objects than SOTA methods,BDNetstill achieves comparable performance.