I live in GA where there is a state income tax. I work in TN where there is no income state tax. My company does not operate a business in GA and are telling me they cannot withhold GA taxes for this reason. Is this accurate?
As indicated by the comments to the question (credit: RonJohn), if the TN company has no GA presence then they may not need to setup GA payroll/income tax withholdings. It may be more of a question of not wanting to deal with the paperwork/red ink/bureaucracy of paying state taxes for a single employee, who could decide to move elsewhere in the future.
Instead, you are most likely responsible for making estimated tax payments to the state of GA, for the amount that would normally be withheld from your gross income.