Narrative visualization transforms data into engaging stories, making complex information accessible to a broad audience. Foundation models, with their advanced capabilities such as natural language processing, content generation, and multimodal integration, hold substantial potential for enriching narrative visualization. Recently, a collection of techniques have been introduced for crafting narrative visualizations based on foundation models from different aspects. We build our survey upon 66 papers to study how foundation models can progressively engage in this process and then propose a reference model categorizing the reviewed literature into four essential phases: Analysis, Narration, Visualization, and Interaction. Furthermore, we identify eight specific tasks (e.g. Insight Extraction and Authoring) where foundation models are applied across these stages to facilitate the creation of visual narratives. Detailed descriptions, related literature, and reflections are presented for each task. To make it a more impactful and informative experience for diverse readers, we discuss key research problems and provide the strengths and weaknesses in each task to guide people in identifying and seizing opportunities while navigating challenges in this field.