I think that - fresh ( can be also green ) leaves can't be burnt - is working better :)
In my 58 years of life I have never heard anyone say in any variety of English that something is "unburnable"!!!! Yes something cannot be burnt, is not combustable, is flame resistant......
I am a native English speaker and I would never say "unburnable". I can see what it might mean, but here I would probably say "Green leaves don't burn well."