Although I haven't taken the RS tour of Turkey, I have been to Turkey 6 times, and I think I have been to pretty much every place on the RS itinerary.
Unless you have trouble with weak ankles, I wouldn't worry with hiking boots. I have never taken anything heavier than Merrell Jungle Mocs. There have been a few occasions -- hiking into the Taurus mountains from Aga Limani or hiking portions of Termessos -- where I would have appreciated some light hiking boots. However, neither of those places is on the RS tour.
You generally don't need hiking boots for the ancient sites -- Aspendos, Aphrodisias, Ephesus, Hierapolis, and such. If you hike the travertines at Pamukkale, you have to do it barefoot --- no shoes or socks are allowed. I am not sure how much hiking is involved in the Cappadocia part of the tour. I have only been to Goreme National Park and one of the underground cities in Cappadocia, and hiking boots aren't necessary for either place.