Finishing drawers on outdoor furniture

I am finishing up a teak sideboard thing with granite top and sink that will sit next to our outdoor grill. Planning to do an awning at some point, but for right now it will be exposed to rain.
There are two drawers, one on either side of the sink. The drawer boxes are white oak, with teak false fronts. The drawer bottoms will be loose-fitting Starboard plastic lumber pieces resting on small cleats at each corner, possibly at a slight slant, so they shouldn't hold water if any rain gets in past the granite overhang, and I can just pull the Starboard pieces out and put in the dishwasher if they get dirty.
Trying to decide if the white oak drawer boxes need any kind of finish, mainly to protect against mildew or make them easier to clean if mildew develops.
Not really concerned with a lingering oil-based scent, as they are outdoors, but thought a penetrating oil might be better than a varnish as it wouldn't start peeling off after a few months. Or maybe better to just leave unfinished? Anyone have any suggestions (other than not building outdoor furniture with drawers in the first place)? Thanks.

I'd certainly put finish on the oak. I'd worry that at some point some rain will get in to the drawer. It will stain unfinished oak, but it will just dribble off finished oak.

Finish them. If you don't they will start to grow mildew etc. Are you putting oil on the teak? If so, use the same product. Cheers

Was going to try Watco teak oil and reapply every so often.

I would use Waterlox Marine finish.