How do I get this looped animation to stop?

I'd load the Animation outside of the anonymous function's scope.

Mouse = game.Players.LocalPlayer:GetMouse()

Prone = false
Player = game.Players.LocalPlayer.Character

Animation = Player.Humanoid:LoadAnimation(script.Lie)

Mouse.KeyUp:connect(function(Key)
	print("Key "..Key.." down.")
	if Key == "c" then
		print("Prone key down.")
		if Prone == false then
			print("Prone is false.")
	   	         Prone = true
			Animation:Play()
			Player.Humanoid.CameraOffset = Vector3.new(0,-4,-1.5)
		else
			print("Prone is true.")
			Animation:Stop()
			--Animation = Player.Humanoid:LoadAnimation(script.Animation)
			--Animation:Play()
			Player.Humanoid.CameraOffset = Vector3.new(0,0,0)
			Prone = false
		end
	end
end)

The LoadAnimation method creates a packet of information that is stored inside of the humanoid that will not go away.

It's not a good idea to put it locally inside of the function, because every time a key is up, you're loading more packets information.

Yes, it won't do anything because loading an animation will make the animation dormant. But since you're always playing it within the function, every animation you've loaded will be played, therefore making it impossible for the animations to stop.

That's why I made the Animation variable a static one, rather than a local one.