The intrinsic non-stationarity of urban spatiotemporal (ST) streams, particularly unique distribution shifts that evolve over time, poses substantial challenges for accurate urban ST forecasting. Existing works often overlook these dynamic shifts, limiting their ability to adapt to evolving trends effectively. To address this challenge, we propose DOL, a novel Distribution-aware Online Learning framework designed to handle the unique shifts in urban ST streams. DOL introduces a streaming update mechanism that leverages streaming memories to strategically adapt to gradual distribution shifts. By aligning network updates with these shifts, DOL avoids unnecessary updates, reducing computational overhead while improving prediction accuracy. DOL also incorporates an adaptive spatiotemporal network with a location-specific learner, enabling it to handle diverse urban distribution shifts across locations. Experimental results on four real-world datasets confirm DOL's superiority over state-of-the-art models. The source code is available at https://github.com/cwang-nus/DOL.