I live in Florida, I work remotly from my house for a company in Georgia. Do I need to pay Georgia state tax?

If you live and work in FL, then you have no obligation to pay any GA tax. It does not matter that the company may be headquartered in GA - the only things that matter here are where you live, and where you work. If your employer has been withholding GA state income tax - have him stop immediately. If you had GA tax withheld for 2009, you will have to file a GA non-resident income tax return, to document that you have $0 GA-based income, and you should get a full refund.

If the OP files a non-resident return requesting a FULL refund, he/she will have to get a letter from the company which clearly states that she neither worked nor lived in Georgia.

Otherwise, Georgia (nor any other state with an income tax) will NOT believe her when he/she says he/she neither lived nor worked in Georgia.