I've been trying not pressing my binding for the last year since I joined this forum, it was one of my try something new attempts. I'm sort of meh on the results. I typically use a rather full poly batt and I consider the binding the last chance to add fabric to a quilt so use a slightly larger binding cut than many, 3".
Anyway, I haven't found any issues with the crease that some people seem to have when pressing. I didn't have any issues turning my corners or other concerns I had about it being non-pressed although I feel that it is easier to distort/add in fullness with it unpressed. Ultimately, my tops may wait some time before they are finished and I like storing the bindings pressed better for no particular reason other than they look more done. As I mentioned before, I have pieces of cardboard about the size of a paper tablet that I wrap my binding on, it stores flat and well and I can count the wraps and times them by the cardboard size. Don't seem to have problem with the folds, I used to put it around an empty oatmeal carton but that shape takes up a lot of space.