There is no Bible story that explicitly mentions a "burning rock." However, there is a story where fire comes out of a rock when struck. This is found in the Book of Judges chapter 6.
In this story, Gideon is visited by the Angel of the Lord. Gideon prepares an offering of meat and unleavened bread. The Angel of the Lord tells him to place the offering on a rock and pour broth over it. Then, the Angel touches the offering with the end of his staff, and fire flared from the rock, consuming the meat and bread.
So, while not a "burning rock" per se, the closest thing to that description would be the story of Gideon and the Angel of the Lord, where fire came out of a rock.