The borders themselves use slightly LESS fabric if you don't piece them, because you don't use up fabric in the seam allowances of the piecing. But if you are buying fabric specifically for the borders, you can often buy less if you are willing to piece them. Also, if you already have the fabric, you may have to piece it if it isn't long enough. That's why you will hear that piecing the borders saves fabric.
The quilt I'm working on now has an inner border that I had to piece, because the piece of fabric in my stash was not long enough. (I think it was 27" long, and the quilt was around 84" square.) There are at least 2 seams on each side, but I'm confident that they won't show up after the quilt is quilted. If I had not been willing to piece it, I would have had to buy almost 2.5 yards of fabric, and I'm trying to use up stash instead!