I live and work in Florida,but my wife works in Florida and lives part time in Georgia. Do we have to pay Georgia state taxes?

You need to define a bit better what you mean by "my wife ... lives part time in GA." Do you mean she was a GA resident for part of the year but now is a FL resident? Or do you mean she lives most of the time in GA but spends an occasional weekend or two in FL? The issue is to figure out whether GA would think she is truly a GA resident, and that requires understanding where she spends most of her time.

If your full-time residence is clearly in Florida, you have established domicile in Florida, so you should NOT be liable for Georgia income taxes on any earned income.