The fire rainbow is the rarest of all naturally occurring phenomena. The clouds must be cirrus and at an altitude of 20,000 feet at least. There must be just the right amount of ice crystals present, as well.
The sun has to hit the clouds at exactly 58 degrees. It makes the rainbow appear to be on fire, hence the name. It’s actually cold as ice, though. In the weather world, the phenomena is known as a circumhorizontal arc.
It isn’t a traditional rainbow, per se, but an effect that happens when light passes through wispy cirrus clouds at high altitudes. Even more specifically, the hexagonal ice crystals that make up the cirrus clouds must be shaped like thick plates with their faces parallel to the ground for the fire rainbow to appear.
The light enters through a vertical side face of the hexagon and leaves through the bottom causing the light to refract, or bend, like in a prism. If the crystals are aligned precisely, then the whole cloud lights up as a rainbow.