This paper focuses on the AI-based damage assessment (ADA) applications that leverage state-of-the-art AI techniques to automatically assess the disaster damage severity using online social media imagery data, which aligns well with the ''disaster risk reduction'' target under United Nations' Sustainable Development Goals (UN SDGs). This paper studies an ADA model generality problem where the objective is to address the limitation of current ADA solutions that are often optimized only for a single disaster event and lack the generality to provide accurate performance across different disaster events. To address this limitation, we work with domain experts and local community stakeholders in disaster response to develop CollabGeneral, a subjective logic-driven crowd-AI collaborative learning framework that integrates AI and crowdsourced human intelligence into a principled learning framework to address the ADA model generality problem. Extensive experiments on four real-world ADA datasets demonstrate that CollabGeneral consistently outperforms the state-of-the-art baselines by significantly improving the ADA model generality across different disasters.