"Fire Rainbows" are actually neither fire, nor rainbows. They are called so because of their bright colors and flame like appearance. 

They are known as a Circumhorizontal Arc, which is an ice halo formed by hexagonal plate-shaped ice crystals located in high level cirrus clouds.

Bright colored Circumhorizontal Arc occur mostly during the summer. When the sun is high in the sky, sunlight hits flat, hexagonal shaped ice crystals and gets split into individual colors like a prism.

The sun has to be at an elevation of 58 degrees or greater, there must be high altitude cirrus clouds and sunlight has to enter the ice crystals at a specific angle. This is why it's such a rare phenomenon.