An interesting question! No, the effect you mentioned would not, in fact, happen. What is more likely to happen (and what happened on Earth) is for the plants to leech oxygen from the atmosphere, and to... I forget the word, but essentially lock carbon into the ground upon burning. (Burnt plant material!) That process is theoretically where all of our oil originated.

It seems unlikely that a plant would evolve a seed design that inherently lends itself well to bursting into flame in the first place. I'm not sure such a design would last long enough to develop fire-resistant properties, or that an already fire-resistant system would morph into a design that is inherently more susceptible to bursting into flame.

Bark cloth would be exceedingly impractical, as barks would really be more like thick rough plantish shells than bark. Definitely not very conducive to fibers.

Much like the animals, plants that would lend themselves well to being turned into spinnable fibers would be in the cold regions. I suppose it's possible they could have a linen equivalent, then...