Not really, of course, but it's easy to see how that description was applied to an AP photo that's been circulating on the web since June, and recently in some e-mail attachments that have been making their way around as well. Here is a copy of the much-circulated photograph that accompanies the e-mail, and it is indeed a beautiful sight, so much so that some people assume the image is a fake. While it is possible that the picture was taken with a polarizing filter to enrich the sky color, or may have had it's saturation and contrast increased a bit in post-processing, the atmospheric phenomena that is pictured here is quite capable under the right conditions of producing stunningly pure colors in an otherwise blue, gray and white sky.
That phenomenon would be something called the "circumhorizontal arc," sometimes called the "circumhorizon" arc instead, and seems to have only picked up the name "fire rainbow" in relation to the picture above, which was taken during an especially bright and long-lasting example of the type that occurred for about an hour in the vicinity of Spokane WA and western Idaho back on June 3 of this year. One reason the fire description seems apt here is the fact that the cirrus clouds that produced the colors happened to have been arranged in the form of nice fibrous-looking fall streak filaments that added to the impression of flames in the sky. Some other images of the same phenomenon, while gorgeous in their own right, don't give quite the same impression because the ice crstals that make up the cloud are not arranged in the same manner.
Regardless of the large-scale structure of the clouds involved, there are some fairly strict requirements that have to be met for a good example of this display to make an appearance, and for that reason they are very rare. A German study indicated, for example, that on average the fairly common parhelia (sun dogs) are seen about 730 times for every single appearance of a circumhorizontal arc. This makes me feel especially lucky that I saw one of these, and a fairly nice on at that, near the Pamlico River in eastern NC back in the late 1980s. I took a photo of it, and if I can dig it out of whatever binder or box it happens to be stored in at home, I'll make a scan and add it to this post - wish me luck!
As for the requirements for one of these to form, they include that the cloud must be thin enough for sunlight to pass through but thick enough to have a noticeable effect, it must be made up in large percentage of hexagonal plate-shaped ice crystals that are not too small, are well formed with "clean" sides and flat faces, and the majority must be aligned such that their flat faces are parallel to the ground. If those conditions are satisfied, you still have to have the cloud located in the correct portion of the sky. For us, that means within about 40 to 50 degrees of azimuth to either side of a point in the southern sky that is located about 46 degrees of elevation below the sun. On top of that, the sun itself has to be 58 degrees or more above the horizon (here in central NC, that doesn't happen at all between about 13 September and 29 March, so a circumhorizontal arc around here is mostly a sumertime event). Finally, if all that stuff comes together just right and there are thick clouds either above the cirrus in question that block sunlight from reaching them, or there are lower clouds that interfere with a view from the ground, we're still out of luck.
There is some good news here, though, because a closely related phenomenon called the "circumzenithal arc" occurs over 100 times more frequently than the circumzenithal arc, and while still not extremely common, produce similarly brilliant colors in the sky. While the circumhorizontal arc involves sunlight that passes into the crystal plates through the side and is refracted out through the bottom of the crystal, the circumzenithal arc is produced by very similar cloud formations, but involves light passing into the top face of the crystals and being refracted out through the sides. While in both cases the crystals effectively act as tiny 90-degree prisms, this seemingly small difference means that for the circumzenithal arc to be seen, the sun has to be at or BELOW 32 degrees above the horizon, which can occur anytime of year during the morning or afternoon. This means a good part of the day in winter, since our maximum mid-day solar elevation near the winter solstice is only about 31 degrees. When conditions are right, these arcs are seen well ABOVE the sun, in the form of an arc that is seen by looking in the direction of the sun but almost straight overhead, in which case a well-formed arc will seem to form a colorful "smile" of sorts in the sky.
There is a very nice web site that covers a wealth of atmospheric optical phenomena that has some good images and diagrams further explaning these arcs, and many other cool things you might see in the sky. The home page is here. An explanation of the circumhorizontal arc and a gallery of images are located at this link and this one respectively. Here are similar links for the circumzenithal arc and a gallery of example photos.
Finally here's one last link to another nice circumorizontal arc image featured at spaceweather.com. Keep looking up!