With the advancement of 5G technology, Internet of Things (IoT) devices are widely deployed for automated services. However, due to the lack of robust encryption, these devices have become prime targets for network attacks. IoT device identification is crucial for information security, enabling the recognition, classification, and management of diverse devices within the network. Conventional methods, when encountering new devices, require re-engineering features and retraining classifiers from scratch, which is resource-intensive and lacks timeliness. Existing replay-based incremental learning (IL) methods struggle with the class imbalance between new and existing classes in IoT device identification, leading to catastrophic forgetting. To address this issue, this paper introduces IoT-ILDI, a novel IL framework with two main mechanisms: a class balanced fine-tuning mechanism to eliminate training imbalances and an adaptive loss weight strategy to minimize forgetting of previously learned knowledge. The experimental results demonstrate that IoT-ILDI performs well in the incremental process, and the classification accuracy and F1-score on the CIC-IoT-2022 dataset are 98.85% and 95.96%, respectively. Additionally, with limited memory and an increasing number of new classes, IoT-ILDI outperforms state-of-the-art methods at each incremental phase.