It's not the full vertical stabilizer that is painted as the Aircraft rolls down the line, it's just the rudder. If an airline has a fairly simple paint scheme and the rudder is all one colour, it may not be noticeable that it is even painted.
As mentioned by Jetdoctor, the rudder is a flight control surface and has to be perfectly balanced for it to operate correctly. The hinge line of the rudder is held horizontally in a tool (after painting) and if it doesn't lay level, then counter weights located forward of the hinge line are either added or subtracted until it does lay level. It's a very precise procedure, so much so that placing a penny on the surface of the rudder while it is in the tool and moving it an inch forward or aft can change the balance of the rudder.
I used to do this procedure on the BAe Hawk military trainer in production, the same principle applies to large commercial aircraft. Since the counterweights are close to the hinge line they are made from very dense material (I forget the name of the exact alloy, but it's something that when you pick one of these up, you think "Wow, heavier than I thought"). One of the few applications when heavy material is prefered in aircraft manufacture. The same kind of counterbalances can also be found in other locations on various flight controls systems depending on the a/c type and system involved.
Ailerons also require the same kind of balancing procedure before installation.
Also after repairs to the structure in service, these flight control surfaces should be re-balanced.