There is very limited information available what GM's actually do during their workday.
Taken from Main Menu > Support Desk > GM Service Main Menu:

Report a Real Money Trader (RMT) > "we're sorry for inconvenience. thank you." no feedback if any of our reports were heard or if they fall on deaf ears.
Report use of third party software (cheats) > "we're sorry for inconvenience. thank you. use the website because we can't do anything with it."
Report inappropriate behavior > "we're sorry for inconvenience. thank you." the stalker or harasser MIGHT get slap on wrist and 3 day suspension.
Report Violation of Agreement between players > I would love to see example of this. Do they make gil from thin air and give it to someone as compensation for scam? Any stories?
One time item restoration > one time per account meaning each time someone asks for help, another request of the same type will be denied. meaning less and less work over time.

what else?