Question: In my world, I'd like to place a region of permanent forest fire, but I'm having trouble coming up with the minute details of the biome.

For example, my first idea was that it started when a lush, humid forest was exposed to a volcanic eruption, followed by a permanent but slow lava flow, lasting long enough that some plants have adapted and evolved into flora that relies on the constant fire and is itself so slow to burn it can outgrow whatever is lost to the fire.But how plausible is that, biologically speaking?

What is a better approach to creating a permanently burning forest?

Answer: 
Quite a few plants - mostly trees - have evolved to [require fire to germinate](https://www.britannica.com/list/5-amazing-adaptations-of-pyrophytic-plants)! This seems like the seed of what you might need - combined with a geography that's appropriate.

Annual fires are common in several areas; but what makes them not permanent is the weather cycles. This could possibly change, though, and become permanent in a way - if your geography allows the fire to "travel".

Imagine a Pangea type world, a single large continent taking up 30% of the surface area of the globe, say. If that Pangaea is ringed by forest, and if the weather patterns comply, it's possible that you could have a *travelling fire*, which, say, is on the east from March to June, the north from June to September (northern summer), the west from September to December, and the south from December to March (southern summer). (West/East could easily switch, of course.)

The fire could literally travel in that way, ringing the globe every year. Then, cultures would possibly come up that would worship the "renewing fire", have rituals based on it, etc.

I'm not sure what you'd need for the weather to work out this way (you'd need the rainy season to not coincide with the fire, but to coincide with the *end* of the fire, for example), so that part might be a bit handwavy - but it seems like a possibility, if you're not going into meteological science fiction, at least!
You can't. It's simply a matter of energy input and output. The sun delivers about 700 W/m^2 to the Earth's surface. Plants convert a small fraction of that into biomass. (Photosynthesis is at best only about 5% efficient: <https://en.wikipedia.org/wiki/Photosynthetic_efficiency#Typical_efficiencies> )

A fire will convert many years worth of plant-produced biomass into CO2 & H2O in minutes to hours\*, releasing the stored energy in the process. Then no more fire until the plants have a chance to regrow.

Even the coal seam & natural gas fires mentioned in another answer demonstrate this. It took probably millions of years of plant growth to produce that coal or natural gas, which is now burning at a rate far greater than production.

\*Generally speaking. You can have situations, like tree roots, where there's limited access to oxygen, so the fire might smoulder for months.