Overload energies add to the weapon damage potential. It's a 2:1 ratio past the greenbar. (50% of additional damage bonus for max of 100% additiontional energy for overloading).
if the weapon does 10 points of damage at regular load using 10 points of energy. (the green bar) then its Fully Overloaded (red bar) shot does 15 damage points using 20 points. A partial overload is similiar: 15 points does 12.5 points of damage.
(just in case you want to know, underload is 1:1 ratio... Even 1% of energy short of the greenbar reduces the damage potential by 1%. To use the example above, 5 points of energy does 5 points of damage.)
I think this applies for both phaser and heavy weapons.