The short answer is no, but it is highly recommended especially if you can’t supervise them all day. Dogs will lick, bite, and scratch their stitches. A cone will prevent them from reaching these areas.
If you are actively supervising your dog and you can confidently prevent them from giving their stitches attention, then you can leave the cone off. Just remember to put it back on if you’re going to leave them alone, even for a minute.