1. Cut down the fig tree to improve the grass.
2. Leave the fig tree alone and remove the affected grass.

The choice is yours.

Previous answer short and sweet, but absolutely accurate. Why not cut out some grass where you know it will be covered by the fig leaves and plant Lamium maculatum 'White Nancy' or 'Beacon Silver' in it - tolerates shade and dry conditions really well, and won't mind being covered by the leaves by midsummer.

The horse has bolted. When planting a fig, the roots must be restrained by digging a large hole and lining it with flagstones, and filling the bottom with rubble. I would clear the grass a little way from the trunk and perhaps grow some Heucheras, Pachysandras or Vinca minor. When the tree is dormant you could cut down into the soil with a spade about 3-4 feet from the trunk to prune some of the surface roots.