Extreme weather events driven by climate change, such as wildfires, floods, and heatwaves, prompt significant public reactions on social media platforms. Analyzing the sentiment expressed in these online discussions can offer valuable insights into public perception, inform policy decisions, and enhance emergency responses. Although sentiment analysis has been widely studied in various fields, its specific application to climate-induced events, particularly in real-time, high-impact situations like the 2025 Los Angeles forest fires, remains underexplored. In this survey, we thoroughly examine the methods, datasets, challenges, and ethical considerations related to sentiment analysis of social media content concerning weather and climate change events. We present a detailed taxonomy of approaches, ranging from lexicon-based and machine learning models to the latest strategies driven by large language models (LLMs). Additionally, we discuss data collection and annotation techniques, including weak supervision and real-time event tracking. Finally, we highlight several open problems, such as misinformation detection, multimodal sentiment extraction, and model alignment with human values. Our goal is to guide researchers and practitioners in effectively understanding sentiment during the climate crisis era.