It's got nothing to do with winning the state. It's got to do with explaining why you wouldn't prove you are entitled to be on the ballot.
If people are told that all he has to do to show up on the ballot is show an original birth certificate from Hawaii, but he refused to do it, they simply won't understand it. It doesn't pass the smell test. People will think something is seriously wrong, and perhaps he *ISN'T* eligible to run for office.
It will engender a WTF? moment among the people of the United States, and Obama will HAVE to answer their questions. If he can't answer correctly, he will not be allowed to run in ANY state, and it might be possible to revoke everything he has done. (If a Federal Judge ORDERS Hawaii to release the original document and it doesn't establish that he is actually born in the United States.) Keeping him off the Ballot in Georgia will cause an avalanche of inquiry into his eligibility across the country.
Also, the Birth Certificate that he provides must show someone else for a father.