It's worth remembering that Greek fire wasn't a wonder weapon. The Byzantines still lost naval battles even though they had it. It was also limited in that it could only be deployed in relatively calm seas and with favorable winds. That said, there's always the possibility of Greek fire deciding the outcome of a battle and when the butterfly flaps its wings...
Minchandre said:
I suspect what you'll see instead are all sorts of lacquers and (al)chemical treatments to make wood less flammable.
That's a good point. The Muslims found that raw animal hides soaked in vinegar were an effective means of protection against Greek fire. It's logical to assume that the natural back and forth between more effective weapons and defences would lead to the use of chemical treatments to prevent ships' hulls from catching.
I think the biggest change from surviving Greek fire would be in increasing the emphasis on ranged comat. Greek fire would force fleets to keep their distance from each other. So we might see the use of catapults, balista, and the like on ships and potentially an earlier introductiion of cannon.