I would certainly finish the crib with something. It will take a lot of abuse, particularly if it's used by more than one child, and some of that abuse will be (as you note) chewing/etc., which will introduce moisture and other damaging things. It also will be more vulnerable to splintering, which will be painful for your child if it occurs.
The main thing I look for in a crib finish:
- Low or no VOCs; your child will sleep in it, and you don't want carcinogens in the air where you can avoid them.
- Non-toxic; many are labeled this way on the can.
- Non-chipping types of finishes, so either stains or similar that will primarily go into the wood rather than lying on top.
Naturally derived oils are a good example of this; walnut oil, linseed oil, hemp oil, etc., are good choices. Waxes like beeswax are good also, but may need to be re-applied periodically.
We used a combination of linseed oil and beeswax on our child's furniture, and it seemed to work well so far.