As climate extreme and disaster events become more frequent and intense, Geospatial Artificial Intelligence (GeoAI) has emerged as a transformative approach for large-scale disaster mapping and risk reduction. However, the purely mechanical, performance-driven deployment of GeoAI models can result in amplifying inherent spatial inequalities, preventing effective emergency decision-making, and producing severe environmental carbon footprint. To unbox the concept of responsible GeoAI, this position paper examines its emerging role, e.g., in climate extreme and disaster mapping, from a critical GIS perspective. We address the nexus of responsible GeoAI into four interrelated theoretical dimensions, specifically Representativeness, Explainability, Sustainability, and Ethics, with examples from climate extreme and disaster mapping. Moreover, targeting at the operational practice, we then propose a conceptual governance Model of responsible GeoAI that categorizes its governance practices into Data, Application, and Society scopes. Last, this position paper aims to raise the attention in the broader GIS community that the future of climate resilience relies not just on building better algorithms, but on fostering a governance ecosystem where GeoAI is deployed responsibly, ethically, and sustainably.