How to slowly render a line through multiple points in a list?

I have a game object which moves and I constantly store its position in a vector3 list. Later, if I click on a button I am able to get the path that the game object has traversed along the co-ordinates stored in the list. But I need the path to slowly render from the starting point and along the other points and stop at the last point rather than just show up at once when the button is pressed. Any help on this will be greatly appreciated. Please feel free to post any links or code snippets.

Regards, Rahul