Steve, Jim,

I'm a master cabinet maker , and my advice is to finish it inside with the
same type of finish you plan on using for the outside . The reason for
that is to control moisture absorption . With the same type of finish in
and out , the wood absorbs equally on both surfaces , and maintains
balanced internal stresses . Leave a raw piece of wood on a (very) wet
surface and you'll understand ...

And for a machinist tool box, what would you personally recommend for a
finish?

That's the real question. I would rather destroy a box I made largely
for practice than do any damage to my collets. FWIW, the box that came
with my starter endmill set has some kind of varnish on the outside and
bare wood inside. I lean toward oil finish, but have some concerns
about it or anything more modern being in contact with oily collets.

Bill