The company that runs burning man is unlikely to pay for a band; they tend to pay for the plastic arts. Read up on the grant process. You'll find a link on the burningman.com website.
It's a "no commerce" event. (Or before the nitpickers come out of the woodwork, an almost no commerce event.) You wont be able to put out a tip jar or be sponcered by a commercial entity under their banner or anything like that. If some eccentric millionaire choses to pay your way in exchange for x performances that's your business, so long as you keep quiet about it.
I'm sure others will have more to add, but the best plan is to go for the love of it and perform for the love of it.
The Lady with a Lamprey
"The powerful are exploiting people, art and ideas, and this leads to us plebes debating how to best ration ice.
Man, no wonder they always win....." Lonesomebri