The answer is, it depends.
Framing lumber is primarily what would be exposed to rain because that's what the house is before the (rough) roof is put on. In my part of the US, framing lumber is primarily green, not kiln dried. I believe this is purposeful, because it's more pliable before it's fully dried and any lack of straightness can be largely "fixed" by proper fastening. It's also cheaper though, so that could be a contributing factor.
Anyway, green lumber in a house frame isn't going to be significantly damaged by some rain, and, as you say, work would just resume when the rain stopped.
Kiln-dried lumber, on the other hand, isn't going to stay that way for long if it spends any time in the rain. Also, wood stacked in a pile is not going to dry out as quickly as sticks in a frame that are exposed on all sides. So a tarp over the wood pile if there's any chance of rain is a good practice, whatever kind of wood it is.
As far as plywood sheathing and sub-roofing, maybe someone else could weigh in, but exterior grade as it is would definitely stand a couple rainy days without complaint.