Mineral oil should be safe. I use it on my cutting boards, and the maple butcher block table top that I have. You might want to check with a bird source, though I would be surprised if there is a problem. I have a parrot, and I have used mineral oil on the perch. I think you have more problem if the bird chews on the wood. The smell alone might make him mad, but probably not hurt. Come to think of it, we have sliced fruits for "Capt'n Jack" on a freshly mineral oiled board and there have been no problems. His only complaint was me not slicing fast enough. I have never used mineral oil on drawer joints or veneer surface, so others will have to comment on that. Titebond or Elmer's Carpenters glue does not seem to be bothered by mineral oil.
Hope that helps,
maybe butcher block oil? it is safe to use on countertops on which human food is prepared. if not, there are many different oils which are safe for wood on which food is prepared. unfortunately, i am not too familiar with the needs of birds though
I had a few Boos type (that is a brand) butcher block table tops ( 3' x 6' ) for food service. Butcher block oil is fine, but more expensive. Butcher block oil has mineral oil as it's base ingredient, but adds some body agents to thicken, also most types do not change the wood color and absorb a little faster. Not major changes, but big difference in price. Most restaurants and bakeries use mineral oil, unquestionably safe for humans.