After spending some time reading and experimenting I have gone ahead and coded up an old palette based fire effect in Javascript. For those not familiar, the idea is basically to seed the bottom of your image with blocks of "hot" pixels and then let this heat radiate upward. After seeding the image you move pixel by pixel through the image and determine the temperature of a pixel by the temperature of the pixels below it and applying some amount of decay. This essentially allows the heat to spread upwards, cooling off as it does so. Now that you have a temperature for each pixel in the image you simply assign a color for each discrete temperature.
The result is something like this:
Of course, if you don't have a browser that supports the canvas element then this little experiment won't work for you. Also of note - I tested this out in Safari on a PC and the performance was absolutely dreadful; Chrome and Firefox handled it much better. It seemed like in Safari it would accumulate several updates before actually displaying any changes in the canvas, so it may be that there is some other way for me to accomplish this that would work better in Safari.
A fun little experiment that certainly brought me back to my early days of coding. When I am a little less slammed with work I will probably throw together a couple more things like this before moving on to something a little more complicated - maybe a simple game?