I haul wood back when hunting in the Edson area every fall. It's getting harder to find, an access easily. Was using my quad to haul it to the road last year.
Technically you need a firewood permit as well, $5 and available online, good for 30 days and 5 cubic meters of wood.
Personally I'm not a big fan of birch, seems like you can only find green wood making it heavy to pack, and then have to leave it for 2 years to dry prior to burning. I'd rather take dead standing spruce or pine, so much less work.