Addressing the challenges of irregularity and concept drift in streaming time series is crucial for real-world predictive modelling. Previous studies in time series continual learning often propose models that require buffering long sequences, potentially restricting the responsiveness of the inference system. Moreover, these models are typically designed for regularly sampled data, an unrealistic assumption in real-world scenarios. This paper introduces ODEStream, a novel buffer-free continual learning framework that incorporates a temporal isolation layer to capture temporal dependencies within the data. Simultaneously, it leverages the capability of neural ordinary differential equations to process irregular sequences and generate a continuous data representation, enabling seamless adaptation to changing dynamics in a data streaming scenario. Our approach focuses on learning how the dynamics and distribution of historical data change over time, facilitating direct processing of streaming sequences. Evaluations on benchmark real-world datasets demonstrate that ODEStream outperforms the state-of-the-art online learning and streaming analysis baseline models, providing accurate predictions over extended periods while minimising performance degradation over time by learning how the sequence dynamics change. The implementation of ODEStream is available at: https://github.com/FtoonAbushaqra/ODEStream.git.