I don't think they do....at least not without wheel pants. I don't think the ones with fairings do either. Even with the pants on I can feel them shake after take off but a quick tap on the brakes makes it stop. If they were turning they'd be shaking.....maybe?
I have never seen the tires spin in flight except right after take off. The only time I have seen a tire rotate in flight is when ice builds on the front of the tire. After a while, the ice will become heavy enough to rotate the tire with the ice on the bottom.
If you think about it, the fore-to-aft force of the wind on the tire would be equal on the top and on the bottom of the tire and therefore would be equal and opposite spinning force on the tire. Unless the force becomes unbalanced (ice, debris, etc), the tire would not spin.
Look out the window next time you're up. The 172s I flew as a student didn't rotate. Once a tire had a very visible grease mark; on departure, I could easily see rotation stop around treetop high.