Scorsese: He's constantly testing himself, pushing himself. Putting his fingers on the flame is just one thing but then going through the desert and going back out; what is he trying to prove? What is he fighting about himself?
Now, even though I haven't watched the movie from beginning to end in well over a decade, I have enough memory of it to go back and think about it and...yeah, Marty's absolutely right.
Lawrence is always pushing himself to dangerous limits. He does it at the beginning of the movie when we see him going incredibly fast on a motorbike that will eventually lead to his death (The film is told in flashback).
So was Lawrence really trying to prove anything to himself or to others? Was he battling an inner demon?
I have a theory on it and it's not one I've ever shared before.
Shortly after the film gets going, Lawrence is assigned by his commanding officer, General Archibald Murray, for a three-month expedition into Arabia to find Prince Feisal. Shortly before dismissing him, Murray makes one final remark:
I think it honestly comes down to that line from Murray, even though we don't see Lawrence's reaction to it. If anything Lawrence seems to be pleased about this opportunity even though those around him are seeing it as an opportunity to be rid of him.
It's not so much that he wants to prove it to Murray but more to everyone around him. Later in the film when he returns to Cairo after helping the Arabs secure the city of Aqaba in Jordan, he finds out Murray's gone and simply says, "There's a step in the right direction."