I remember a gingerbread house recipe that said that warming the butter and molasses was crucial for construction-grade gingerbread because it made the cookies/building parts stiffer and less crumbly, which makes sense to me -- less air and less moisture make for better gingerbread walls but probably not a technique for chewy cookies.
In general, if you melt the butter, you'll end up with a flatter cookie that has spread and is chewier, esp. if you've added molasses to it. At room temperature, if you've beaten it with sugar, you'll have more air and it'll be cakier. But, it depends on the recipe, how much flour, what type of flour, ratio of butter/sugar/flour, etc.