I just replied on another thread about this. I’ve used this site for research purposes: http://www.chinchillas2home.co.uk/safewoods.htm
Trees whose fruit has a pit I’ve read that they are toxic unless completely dried first: http://www.coloradohrs.com/articles/adopt_prepare.asp
I have also read that they are listed at poison control centers as toxic period and no mention of whether it makes a difference if it is dried.
Plain pine at a hardware store like Osprey mentioned is OK. However birch contains a small amount of salicylates which are toxic and could potentially harm an animal, so I really would not use it. Any red woods I would not consider safe, I would liken it to when you get a splinter in your finger from redwood or cedar and how it gets all hot, puffy, red, pussy and stings like heck. There is a substance in there that causes it and so I really would not want my bunny munching on it. I’ll have to dig out some of my plant books to find the actual name of the substance as I didn’t have much luck online with it.