You will need a $25 per person pass to visit the Uluru-Kata Tjuta National Park which covers three days.  For links about this and other regular passes, see Permits and Passes.  The only other place I can see that you would need a permit is the Hay River track. The transit permit to travel through Atnetye Land on the Hay River Track.  Details of permit application from Direct 4WD (not cheap at $220 plus camping fees). 

Only if your chosen route deviates to include going into the Simpson Desert in the South Australian side of the border, you would also need a Desert Parks Pass.  See Parks SA  (currently $102). 

In general, any time you are transiting through Aboriginal Lands on Aboriginal business roads, or deviating from a public access road, it pays to check if any local permits apply.  Sometimes entering a settlement to purchase fuel or supplies is acceptable whereas some communities require a permit to be obtained.  Some areas and roads are closed to tourists. Enquire with the Aboriginal Lands Council for that area, or the state Aboriginal Authority for that state. 

Bear in mind that many Aboriginal communities are dry communities; no alcohol allowed.  However, in most cases for genuine travellers, having a small quantity of unopened alcoholic beverages for your own use packed away whilst transiting is acceptable.   See previous questions on this topicQ27 Far North Queensland and Q11 Central Australia.  Being older questions and answers, some of the links given may no longer work.