I don’t think so, as being colored blue at room temperature and burning blue are different things. If you think about it, when you burn colored paper you don’t get a correspondingly colored flame.

The pigment is an organic molecule so it'd burn the same color as other organic molecules as the fire breaks it down https://en.m.wikipedia.org/wiki/Xylindein. Flame color is based on the elements in the flame. https://en.m.wikipedia.org/wiki/Flame_test

I doubt it. All the blue wood that we saw was very brittle and/or mushy. I think the mushroom probably needs the wood to be somewhat broken down (and maybe soaked?) to start colonizing it. And then it breaks it down even further. But I suppose it might look really cool in some resin projects!

Edit: Look what I just found on Wikipedia:

It is this compound that is responsible for the characteristic bluish-green stain of wood infected by this species, used today in decorative woodworking such as Tunbridge ware and parquetry. The use of this wood, known as "green oak", goes back to 15th century Italy, where it was used in intarsia panels made by Fra Giovanni da Veroni