Pros of turning: More oxygen, so microbes work faster.
Cons of turning: A compost heap is a potential wildlife habitat, with snakes and hedgehogs and the like and turning it will probably upset these.
That is pretty much what I can think of. What you could do is have two piles and put most of the faster composting organic matter in one pile and turn that, and put the slow composting matter like branches and various woody material in the other and let it sit. Seems to me like you would get the best of both worlds so to speak.
cons of turning: You might turn ten times over the period of two months, or maybe 3-5 hours of labor (or more) You built the pile in the first place, taking more labor to do so. You spend time moving the compost back to your field or garden. Pros of turning: Composts faster and at a higher rate of temp which kills some pathogens and weed seeds. In a nice neat "pile"
cons: takes much longer to break down. Maybe as much as a year or more if it is unbalanced. You built the pile in the first place, taking more labor to do so. You spend time moving the compost back to your field or garden. Pros of not turning: You don't do 5 hours of labor to turn it. In a nice neat "Pile"
Alternative: let the stuff you would put into the compost, sit where it lays. It composts there, in place, and reduces the overall labor. You can't compost all the green matter of an acre (thousands of pounds worth) in a pile with a shovel.
So, if you are composting small amounts, a small pile makes some sense. If you are composting all the greenery from a 5 acre hobby farm, the labor would kill you...