Move that mountain over there.

Actually, when I think about it, you sort of can. Because the mountain is a part of Earth you can just move the Earth. As you know, every action has an equal and opposite reaction so you can just start walking or running by which you ever so slightly changed the Earth's rotation and kind of moved the mountain along with it.
But actually, speed is relative. So if you're walking (which the mountain probably isn't, at least not in exactly same direction at exactly same speed) the mountain is moving relative to you.
Although you could also probably just kick it. Vibrations are also movements, no matter how small.

But what if you just took a rock from the mountain and thrown it somewhere else? If it's considered a part of mountain, you just moved it relative to Earth which is a big success in this case. I mean, think about a pile of gravel. Take some from one side, place it on the other, it's still a pile of gravel and it is also slightly re-positioned.

Anyway,...