And how would you extract the oil? Asking it to leave the tree in a non-energy-consuming manner?
As far as i know the most common way to extract birch oil in larger quantities is by steam destilation.
But no matter which method you use it will demand the use of energy. The part of the plant that contains most oil is the leaf buds in the spring.
You could maybe squeese the oil out of the buds the same way they make olive oil. But My guess it would be very labour intensive and time consuming. Just to gather enough buds to extract any usable amount of oil would take ages....
Absolutely a interesting project, but I think I would forgett the eco-friendly low footprint part of it, as I belive the bottom line would end up negative on the hydrocarbon side.