The proliferation of online news poses a challenge to extracting structured timelines from unstructured content. While recent studies have shown that Large Language Models (LLMs) can assist Timeline Summarization (TLS), these approaches primarily treat models as passive generators. The emergence of Large Reasoning Models (LRMs) presents an opportunity to reason over events actively, enabling iterative evidence acquisition, the detection of missing events, and the validation of temporal consistency. To systematically leverage the reasoning capabilities of LRMs, we propose TimelineReasoner, a novel framework that shifts TLS from static generation to an active, reasoning-driven process. Unlike prior work, TimelineReasoner adopts a two-stage framework: Global Cognition, which tracks events at a macroscopic level and continuously updates a global event memory, and Detail Exploration, which identifies informational gaps and refines the timeline via targeted document retrieval. To support this, TimelineReasoner incorporates several specialized mechanisms, including an Event Scraper for retrieving temporal event descriptions, a Timeline Updater for refining the timeline, and a Supervisor for detecting gaps in the timeline and guiding retrieval. Experimental results on open-domain TLS datasets demonstrate that TimelineReasoner significantly outperforms existing LLM-based TLS methods in terms of timeline accuracy, coverage, and coherence. On closed-domain TLS datasets, our method performs on par with or exceeds state-of-the-art approaches. This work not only pushes the boundaries of TLS but also highlights the broader potential of LRM-based reasoning frameworks for timeline summarization.