What's the best way to remove Sharpie ink from a varnished bar top?
A trick I've seen is to use whiteboard markers to draw over the permanent mark, then wipe it up while it is still wet. This is because whiteboard markers and permanent markers use similar solvents. 

If that fails, you can use Xylene (the solvent in these pens) or Goof Off which is a heavy-duty remover. Neither should hurt the bar surface (unlike scrubbing), but you should test it on a less visible surface first.