Do you have to purchase the flame particle texture to use the flame forge?

No, you do not need to purchase the flame particle texture to forge. All you need to do is go ingame and click
esc -> setup -> utils -> Flame forge
And create your own flame. Though depending on your flame's attributes the price to forge the flame varies.

The texture allows you to upload an image so your flame has a texture.

Flames are made of flame particles.
After you get a flame, get a flame particle texture if you want to change one or more of the attributes in the flame.
(I.E. It can be color/size/etc.)