It's an american magnolia and I"m not finding much online. We lost a lot of the tree in the storm and that would be more useful than burning it. It is still fully leafed.
Magnolia officinalis safe taken orally, but extract injected causes anesthesia and possible death, because intestines do not absorb enough of the compounds involved to be harmful: http://books.google.com/books?id=DyIlwr ... ty&f=false
I couldn't view the links but I'd be careful of anything that indicates it is toxic but would pass through the digestive tract before the toxins could be absorbed. Rabbits have a very long and thorough digestive tract and would be more susceptible to this kind of toxin.