Luck only affects procs. Money bin has no procs so it dies not affect it. Things Luke knux or muli or shadow habe procs.
As Kami said, Luck only works for "Chance to" effects (Chance to Stun, Chance to Forge Item, Chance to Instakill, etc). It works as a multiplier of base chance (As in, a 10% Luck bonus on a 10% Chance to effect makes it 11%, not 20%). This can be increased up to a maximum of 80% Chance to. Technically yes, every tower can increase its Luck, but on some towers (Like Hitman or Gib the Frozen Demon) it won't have any effect unless they're using an item that also has a "Chance to" ability (Such as the Withered set).