Putting the tent up won't result in any real wetness inside the tent. They pop open in like 15 seconds.
Putting it away wet will result in a wet mattress, however. It's not soaking wet or anything - the tent fabric only holds so much moisture anyway - but it will be wet. You'll want an hour or more to air out the next night, if you can swing it.
I usually try to wait for a break in the rain, or a let up and then put it away as dry as I can - shaking sides, and whatnot. I've thought about maybe laying out some trash bags to isolate the wet tent fabric from the mattress, but haven't gotten around to trying it out. Out of the 100 or so nights I've been in the tent, rainy mornings have been maybe 10 of them.
It's the downside of having a mattress in your tent. It's a manageable downside, though.