In the 1980s, I borrowed a record from a public library, which told a story about a dragon that could not breathe fire. It would have been a reading of an existing novel that I would like to locate. I think that the record was part of a series of recordings of children's fantasy books that also included The Phantom Tollbooth. At the very least, I recall getting out a record of (part of) The Phantom Tollbooth at around the same time. (After first posting this question, I thought to call the library and ask them about it; however, they do not have lending records extending back that far, nor do they have a record of what LPs they once had in their collection.)
I think that the actual protagonist was a child from our world (although I may be getting it confused with The Phantom Tollbooth on this point), who somehow travelled to the dragon's fantasy kingdom and encountered various characters who taunted the dragon about his inability to breath fire. There was a jailer who threatened the dragon with his chain and, when the dragon could not breath fire, said, "You're not a dragon, but a mouse. Here's your cheese." Then the king showed up with a bunch of courtiers (including "ne'er-do-'ells") who all tauntingly sang, "Breath us fire. Breath us fire. Breath us fire, little dragon!"
Eventually, the kid does manage to get the dragon to breath fire, and this leads to some kind of further adventures. Beside the fire-breathing dragon, which was a forest dragon, there were other dragons, including sky dragons. The protagonist sees a sky dragon breathing clouds at one point.
So what was the book this was based upon?