I have done a little research and have a few thoughts:
1. If you paint, acrylic paint is best for moisture in a hive to "breath" - it's much more permeable than latex and oil based stuff. But is that good or bad as bees need water and comfortable RH in a hive? A little wind and a lot of moisture is lost in a hurry unless it is cold to really cold or a good wind bloc.
2. If moisture and free water is trapped in the wood then there is the issues of "dry rot" - strange description.
3. But, if the internal hive atmosphere is kept warm than vapor pressure drives moisture out - apparently at a rate that can be managed by the bees. High internal temperatures also help the wood to reach maximum moisture content.
4. I do not paint inside a hive box ( and never will) but I do encourage propolis inside by using rough sawn pine - good system it seems, as it is anti-bacterial and breathable while resisting condensed water formation ( gotta work on that one). Pine makes a good moisture buffering material as it can absorb a lot of water, especially at warm temperatures, and give it back . (That warm internal temperature thing keeps coming up!)
5. If my insulation system works year round -

- I will not paint again, maybe - I think. If I do it will certainly be with acrylic paints. BM's stuff is the best I have seen - I have a lot of acrylic painted pine fence boards still going after 20 years - one good coat.
6. I have seen a lot of dry rotted wood encased in oil lead based paints; wooden boat days.