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: 
It's science-based.

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Nope, for each additional 10°C temperature, the speed of chemical reaction roughly doubles. Between, say, 40°C ambient for normal life to a 600°C for a puny fire temperature, you have 256 difference in the chemical reaction speed\*\*. There's too much of a difference for the life forms to be based on biochemistries close enough to evolve one from the other.

\*\* [Citation](https://courses.lumenlearning.com/introchem/chapter/factors-that-affect-reaction-rate/) `Raising the reaction temperature by 10 °C can double or triple the reaction rate. This is due to an increase in the number of particles that have the minimum energy required. The reaction rate decreases with a decrease in temperature.`

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> Is there a better approach to creating a permanently burning forest ?
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Your only chance, by direct evolution in hot conditions.
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.