Simpson Desert

The SA desert parks pass you have to get to do the crossing should give you all the info you need on that and other matters.
I do recall that the last time we were at Dalhousie Springs, fires were banned there, though I think that was more to do with campers cutting down the local vegetation.
On our Simpson crossing, fires were permitted/essential, but very useful to have your own wood supplies (such as some old Ghan sleepers).
Cheers

Fire Bans Simpson Desert SA Parks

Fire information

Wood fires and solid fuel fires are prohibited from 1 November 2015 to 31 March 2016.

Gas fires are permitted other than on days of total fire ban.

Can't download the information, but the Qld NP site says no campfires permitted in Simpson Desert NP, Munga-Thirra NP.

Times have changed or I am just getting old .I lived and worked in Simpson and we always had camp fires ..I can understand now more people cross it therefore wood would be scarce ..

Bring your own wood !! Tis illegal to cut down or pickup even fallen timber , small fauna require the fallen timber for their very survival …..

There is no single answer.

In the Simpson Desert within Qld - MungaThirri, you aren't allowed to collect wood or have fires.

In the Simpson National Park in SA you can have fires out of fire season so April to October is ok.

No firewood collection or fires in the Witjira national park. - Basically west of about Purni Bore, covering Mt Dare, Bloods creek, Dalhousie etc.

As a rule, wood is harder to get the further west you are, and getting harder to get in general.