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: 
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.
**Fire loops around the globe**

Your planet has its landmass as a band around the equator.
(or, alternatively, there's [almost] no open water, poles are covered in snow, leaving only the equatorial zone available to the biomass)

Somehow, there are no large rivers.
(plants still need to get moisture, and there's evaporation, which means there ought to be rain- or snowfall, which usually means runoff, but fires tend not cross large bodies of water... figure this out).

The equatorial zone is dry-ish, and e.g. typically covered by grasses.

The fire started (magically in only one direction), spread the entire width of the equatorial zone, and continues to burn, e.g. westwards.

By the time the fire has reached it's origin, that part has already sprouted new mature (and dry) grass, with seeds either preserved deep in the soil, or spread by wind, or spread by birds or even migratory animals.

Thus the fire continues to burn in a circle around your planet.

The long-lived plants died out, what remains are grasses and maybe shrubs.

The plants have therefore evolved to live in this cycle, and so have the insects and the animals.

P.S. some math

Fire boundary speed: 10% of wind speed (<https://link.springer.com/article/10.1007/s13595-019-0829-8>)

Typical wind speed: 10km/h
(<https://sciencing.com/wind-speeds-tropical-rainforest-23367.html>)

Thus, for a planet the size of the earth, the fire circumnavigates in ~4½ years, leaving enough time for plants to germinate, go through several life cycles and accumulate large amount of stale, dry biomass.