I don't think it's an invasion of privacy unless you go through their personal belongings. Pulling your own towel off your own furniture is fine; opening drawers or closets where you've told them they can store their things is not.
Even in a hotel, the housekeeping staff is going to be picking up towels, making beds, vacuuming, etc., around the guests' things.
I agree with this. If you have a legitimate reason to go in the room *of yours* that they are *borrowing* while they partake of *your hospitality,* there's no need to worry about invasion of privacy. I know that I, as a guest, certainly would think you might go in the room. It's your house, after all.
And I'd mention the water damage in a matter-of-fact way.
(and not, "the chair's old, so the varnish won't protect it," but "since the towel was damp, I was worried it would damage the finish, sort of like a glass without a coaster. It did, a bit--you have to watch out for water, it's really damaging."