A "burning question" is a phrase used to describe a question or concern that someone has that is urgent and important to them. It is a topic or issue that they feel strongly about and want answers or solutions for. Full definition
Any other burning questions about the future of location marketing tech? (inc.com)
Our hand - picked board of advisors answers burning questions on the best strategies for everything from building an online reputation to branding and social media marketing. (entrepreneur.com)
Although, I suppose it would answer that lifelong burning question of whether she wants to know me or not. (slaw.ca)