The setting I have in mind is an Earth-similar planet which has most of its landmass covered in forests. A civil conflict on this planet ingnites forests fires, which then spread. I was wondering if this fire would be able to spread throughout an entire continent if said forest was mostly continuous and there was no fire stopping service available. Is such a continent wide forest even possible?
Continent wide forests are fine, it's just that we've cut them all down apart from the Taiga.
The catch is that ours were temperate or tropical rain forests. In both cases unlikely to have continent wide fires due to being too damp. The Taiga is a snow forest and equally not particularly flammable.
What you need to look out for are two factors. Pyrophytes and climate.
Passive pyrophytes sometimes use fire to germinate their seeds, meaning that if there isn't a fire they can't reproduce. They'll evolve in environments with regular small fires that clear the undergrowth and leave space for new trees to grow. It's these regular small fires that keep the fuel down and stop the fires from spreading too far when a big one does come up.
Active pyrophytes are the more interesting one for your purposes. Australia has the best known of these with the eucalyptus. Eucalyptus oil is highly flammable and is contained in the air around the tree, any spark can cause raging fires.
A continent wide eucalyptus forest, with the right winds and climate, could quite happily rapidly spread a fire from one side to the other.
Under the conditions you propose, of continent-wide forests and a lack of fire-fighting services, then continent-wide forest fires are not only possible, but they are likely to occur regularly.
Suitable weather patterns like prevailing winds across the continent will greatly facilitate any forest fires. With an extensive enough forest any forest fire will probably transform into a firestorm. Once that happens, there will be no stopping it.
In conclusion, continent-wide forest fires are possible.
Unlikely but possible
Forest fires don't spread indefinitely, wind patterns, terrain, and rainfall tend to limit them. Humans fire-fighters often make it worse by stopping fires, creating an unnatural much larger build ups of fuel. naturally fires burn up this fuel periodically, it tends to be self regulating large build ups of fuel are easier to ignite so fire becomes more and more common meaning naturally getting to the m massive amounts of fuel necessary is aloo but impossible.
Of course something like the chicxulub impact triggered continent wide fires just by spraying molten rock particles and burning ash over a huge areas, IE most of North America.
the size of the forest is your real problem, having a continent wide forest is only possible if you have a small continent at the right latitude. there are many biomes that cannot support trees.
This is pretty unlikely, for a few reasons: a) If trees were adaptable to that many terrain types, we'd have fully forested continents in real life
b) If your forest was that flammable, it'd already have gone up in a lightning strike
and c) Eventually a mountain range or very wide river will interrupt the fire.