Hyper-yields, baby!

Check out the food and production yields on those burning woods!
Note: cities can work those (hyper-yielding) burning tiles.

A forest fire that goes on forever.

Forest fires in Apocalypse mode have an approx. 5% chance of spawning.

To achieve a forever burn, you need to plant woods along corridors connecting larger swaths of (old) woods. This is to create a larger mega-forest. The mega-forest will help to perpetuate a burn creating a forever burn.

(I hypothesise fires start on old-woods rather than new or planted woods. Without access to the game code we can't know for sure though). Certainly this strategy of creating a mega forest from the existing "old-woods" will help ensure that naturally occurring fires will happen on your forest.

Ultimately though, getting a forever burn is heavily RNG dependent. Given that, this guide shows you how to keep the fire alive.

• Pick a map with a large swath of woods or rain forest. Real world maps are ideal. E.g: Europe, Earth.
  
• Choose a civ and opponents such that you will spawn next to the forest. E.g: Vietnam, Kongo, Norway, Bull Moose Teddy.
  (Picking AI opponents who do chop help. E.g: Nubia, Gaul, Maya, Ethiopia).
  
• Turn Apocalypse mode on.
  Other modes that work well with this strategy (but non essential) are: Zombie Defence and Dramatic Ages. (Both zombies and dark ages help to save forests from humanity).

Once you have settled the forest or rain-forest, next build a few units to explore the boundaries of your forest and capture neighbouring workers. This is to prevent them from chopping your forest. That includes attacking city-states too. If you are playing Zombie Defence, let the zombies kill those city states and neighbouring cities near your forest.

Pillage for faith.

With 300 faith buy a Soothsayer.

Position the Soothsayer on a grassland or plains woods tile and create a disaster, for a forest fire.

Keep buying Soothsayers (and participating in Appease the Gods competitions) to keep on burning.

(Or as Vietnam beeline Medieval Faires). Then you can plant forest on corridors connecting forests to create a mega-forest and a mega-burn.