Open-Domain Timeline Summarization (Open-TLS) seeks to construct coherent evolutionary narratives for a topic by distilling information from large, unstructured web streams. Existing retrieve-and-generate pipelines often fail to capture logically necessary but semantically distant evidence, resulting in macro-level causal gaps (missing intermediate stages) and micro-level element gaps (underspecified milestones). We propose Active Timeline Completion (ATC), an iterative framework that converts implicit narrative gaps into targeted retrieval questions, enabling reflection-driven evidence acquisition. ATC leverages three complementary mechanisms—parametric causal gap detection, historical trajectory analogy, and dynamic ontology-guided completion—while a Logical Utility Verification module ensures only evidence directly addressing these gaps updates the timeline. Experiments on Open-TLS demonstrate that ATC consistently outperforms strong baselines in both content coverage and temporal alignment, showing that active, question-driven retrieval improves the completeness and coherence of open-domain timelines.