Hyper-yields, baby!

(Check out those food and production yields on those burning woods).

A forest fire that goes on forever.
One that doesn't require Soothsayers to keep it burning. In other words, a forest fire so continuous that Soothsayers are redundant. (For the purpose of starting fires).

What you need:

1. Apocalypse mode.
   
2. Arboreal (or rain forest) map. E.g: Europe.
   
3. Play a civ with a woods start bias. (Or play against civs with non-woods start bias).
   
4. No chop
   It is essential that you do not chop forests. Forests that will fuel the burn.
   
5. Play against civs that do chop. E.g: Nubia, Gaul, Ethiopia & Maya.
   
6. Start a burn asap.
   This requires purchasing a Soothsayer for 300 faith. The easiest way to get 300 faith asap is to pillage holy sites and/or improvements like camps.
   
7. Beeline Conservation, to gain the ability to plant forest.
   (Or if you are playing Vietnam, Medieval Faires).
   
8. Plant forests along corridors connecting larger swaths of forest. This is to create a much larger mega-forest. The mega-forest will help to perpetuate the burn creating a forever burn.

Ultimately though, getting a forever burn is heavily RNG dependent. Given that, this guide shows you how to potentially achieve that. Failing which, obtain a burn that is nearly a forever burn. A forever burning forest (with the aid of Soothsayers 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.
  
• 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. It will certainly be 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.