The lightsaber is shown many times to burn and melt objects.
Such as when Anakin loses his arm against Dooku in Attack of the Clones, you can see his skin and clothes burning.
Or when Darth Maul stabs Qui-Gon you see a burn mark on his clothes in The Phantom Menace
Or When Qui-Gon Melts the Trade Federation ship doors in The Phantom Menace
It may even steam in the rain, as this picture taken from screen shots of Attack of the Clones does show a fuzzier, more jagged looking lightsaber blade when Obi-Wan is in the rain on Kaminio than when he fights Dooku on Geonosis, which could be caused by steam.
And depending on the level of canon we are delving into, the game Jedi Outcast 2: Jedi Academy shows definite sparking from rain on the lightsaber.
The clone wars cartoon shows actual steam coming off of both Anakin and Assajj Ventriss' blades.
Heat is therefore produced one way or another.
This heat would transfer to a flammable material and cause ignition.
Even if the lightsaber isn't hot and "...it just makes atoms vibrate and hence heat is produced in a material the lightsaber comes in contact with" (which I have never heard) when a lightsaber blade came into contact with a flammable or combustible material it would make the atoms vibrate in that material, it would then heat up and cause ignition within itself.
It is still creating heat through one mechanism or another.