There should be no problem with oak wood. The reason oak leaves and bark are spoken ill of is due to the tannic acid in them. It is in the same feeding category as oxalic acid. Most all hardwood leaves have tannic acid in them. In excess it is a problem, the same as Oxalic acid is a problem in some vegetables. Almost every leafy and/or root vegetable we have has oxalic acid. Unless eaten in massive quantities, one only has to be careful in certain ones such as rhubarb.
There are toxins, in everything from spinach to kidney beans, to zucchini. The word "toxin" should not be equated with the word "poison" .... A little oxalic acid or tannic acid is no harm to anything. Even vitamin A and D are toxic in large amounts.
I routinely pick clover to feed my rabbits, this time of year and in the fall, there are usually Oak leaves mixed in. I used to bother taking them out. I have put rabbits outside in a kennel and let them eat what they want. They will almost always select a dry oak leaf or two to eat first. One will not see them eat another, until the next morning when they are put out, when the process will repeat itself. I have observed no harmful effects from this, and the rabbits apparently think the need something from those leaves (they will pick them, over any of the other fall leaves floating around the yard).
With common herbaceous toxins, like oxalic acid and tannic acid, it is basically just a matter of common sense. Rabbits give free range may eat an oak leaf or two. The same rabbits in a cage, might eat enough oak leaves to cause themselves harm if they were offered no other vegetative source.
Oak wood is fine. An oak "branch" may have a lot of tannin laden bark. It is probably best not to give.
I think we need to draw up some kind of list to differentiate between plants with these kinds of toxins and plants like foxglove and nightshade, etc.
