[haroldo]

Aside from possibly burning out your fuel pump, I dont see any other benefit in running out of gas.
Clearly it wont bring bragging rights nor the respect of one's peers.
As far as the battery is concerned, there is nothing special about the car running out of gas, since the car frequently drives with the engine off.
The battery will continue powering the car running (at speeds under 40 MPH and without the need for heavy acceleration) until it needs to be recharged.
Without gas, presumably it will shut down.
I haven't come across anyone in the forums foolish enough to try this stunt, so I can't say for sure what will happen, however.