Continual Learning aims to learn from a stream of tasks, being able to remember at the same time both new and old tasks. While many approaches were proposed for single-class classification, multi-label classification in the continual scenario remains a challenging problem. For the first time, we study multi-label classification in the Domain Incremental Learning scenario. Moreover, we propose an efficient approach that has a logarithmic complexity with regard to the number of tasks, and can be applied also in the Class Incremental Learning scenario. We validate our approach on a real-world multi-label Alarm Forecasting problem from the packaging industry. For the sake of reproducibility, the dataset and the code used for the experiments are publicly available.