General purpose AI, such as ChatGPT, seems to have lowered the barriers for the public to use AI and harness its power. However, the governance and development of AI still remain in the hands of a few, and the pace of development is accelerating without proper assessment of risks. As a first step towards democratic governance and risk assessment of AI, we introduce Particip-AI, a framework to gather current and future AI use cases and their harms and benefits from non-expert public. Our framework allows us to study more nuanced and detailed public opinions on AI through collecting use cases, surfacing diverse harms through risk assessment under alternate scenarios (i.e., developing and not developing a use case), and illuminating tensions over AI development through making a concluding choice on its development. To showcase the promise of our framework towards guiding democratic AI, we gather responses from 295 demographically diverse participants. We find that participants' responses emphasize applications for personal life and society, contrasting with most current AI development's business focus. This shows the value of surfacing diverse harms that are complementary to expert assessments. Furthermore, we found that perceived impact of not developing use cases predicted participants' judgements of whether AI use cases should be developed, and highlighted lay users' concerns of techno-solutionism. We conclude with a discussion on how frameworks like Particip-AI can further guide democratic AI governance and regulation.