I created the first animation by running Script-Fu / Animators / Fire on the head of an arrow. I used an angle of -60 degrees so that the flames go to the left and a partially transparent gradient to preserve the transparency of the background.

I used Filters / Render / Gfig to create the full arrow. The head, body and tail were separate objects so that I could use different brushes and colors for each. I punched two small holes in the body of the arrow so that the fire can be seen through it (and appear as if it was on top of the arrow).

By merging this arrow with the animation, I get an arrow that has its head on fire.

The animation used here is the one that was created above. The background image can be any image that tiles seamlessly (horizontally).

I merged the animation with the background image, and specified a final X offset of -128 (the background is 128 pixels wide). This results in an animation with a nice moving background.

Note that you can also use a background that is wider than the image, as long as you specify the width of the background for the final X offset if you want the final animation to loop.