Our chimney goes right thru the center of the house with the framing almost touching it or touching it in spots. Our chimney, if built when the house was (1890) is also unlined.
My inner temperature of my exhaust pipe, when burning really hot is around 220-270 (max) degrees, the outer is much, much cooler. Maybe 100-120 degrees. The bricks are only slightly warm to the touch, but only at the bottom where the exhaust goes into the chimney, it would gradually get cooler as it goes up.
All cases are different, but that was my concern too! Also, my pipe is really close to my ceiling/First floor rafters, I just put a piece of sheet metal with an air gap and it is cool to the touch.
As you can see, I am less than 6" from the rafters, I have some sheet metal and a old road sign to deflect the heat. This was the only place I could hook it up.
So far been runing for 2 months 24/7.