It is 69%
Just kidding, let's get serious.
I tested this, only took me half an hour.
I tested this 50 times using a Flame Orb, not including the first turn when the Pokémon is inflicted with a burn. Out of those 50, 15 times the Pokémon healed itself and 35 times the Pokémon didn't. So contrary to popular belief, there is probably a lower chance of it healing than not.
Because I did it 50 times, I will times both numbers by two, to get it out of 100:
15 x 2 = 30
35 x 2 = 70
So there probably around a 30% chance that the Pokémon will restore its own status condition, and a 70% chance that it will not.
This is also backed up by the fact that this is the same chance of reeling a Pokémon in with a rod.
If you want me do it 100 times, that's fine, but I'm sure this is close.
Edit: I did it another 50 times; the Pokémon healed itself 11 times and did not 39 times. So if we add this to the original numbers:
11 + 15 = 26
39 + 35 = 74
So the chances are 26% of healing and 74% or not, so the actual chance is probably in between 25-30% for curing the condition.