bathtub in Carolina climate

we live in the states, in North Carolina, and we have weather similar to Japan, i.e. hot and humid in the summer, and cold and dry in winter. What effect does that have on the wood?

Well a climate similar to japan you will need to:
1) in summer: provide a good natural ventilation. Once you empty the tub, wipe it with a dry cloth and leave the windows open.
A wet climate may cause mold stains. On the other hand, this problem is minimized because we use a water repellent. See the documentation on our HP:

2) in winter: avoid using heating equipment in the bathroom when the tub is empty. Use the tub often (at least 2 times a week) and if you leave for an extended period, leave a bucket of water inside the tub and cover it to increase the air relative humidity. Keep an hygrometer near the tub and assure that humidity does not drop below 50%.
If the climate is too dry, there is the risk of the wood drying and eventually cracking.