Since this seems to be a point of discussion, I've thought about it a bit, and I have to modify what I said or implied above: I can recall at least three times when I've run or moved after hearing "fore." I almost always "duck and cover," but in these cases I did not.
All three times I was observing golf being played (you do a lot of that as a college coach). In two of the instances, I ran away and the ball landed where I was standing. In one of those two, I ran under a shelter a few yards away. In the third instance, I ran to the right when I was left of a green, and the ball landed closer to me than it would have. This was not one of my college players.
They know to yell fore if the ball has a chance of hitting me. If I'm near to something that provides cover, I'll sometimes cover my head and get behind it. If I'm out in the open, odds are I'm watching the ball, but if I am not, I'll duck and cover.
I just asked my wife, as she watches my daughter play in events with me, and she said she sometimes runs if she can get behind a thick tree trunk or something, but otherwise ducks/covers. Then she said she usually just hides behind me.
I can see the logic in doing both, but usually, duck/cover wins out. But, still, I have to take back what I said above about how I always duck/cover. I do not. I have also run in the past. It's a little game theory, really: if someone is yelling "fore" the assumption is the ball is going to hit me or be close to hitting me. If I'm in the open, I feel like I duck and cover. If I'm near a thick tree, I'll probably take a step or three and get behind it. (This isn't included in "running"). And there are times, still, as I noted that I'll run: most often when my college kids yell "fore" because we've established a rule there.
Edit: I can see logic in both running or ducking/covering. It depends on your environment. If you can take four steps to get behind a wall or a building or a thick tree, that makes sense to me. When watching golf tournaments, we'll generally see the balls being hit toward us, but when there are trees, we're often standing near them so they can shield us from balls coming toward us. I don't consider it "running" to take a step backward and cover our heads if someone flares a driver toward us.