Is there a way to sync an iPhone 3GS with Mac OS X Tiger?
Yes, upgrading the Mac to Leopard will work. 

Each new iDevice has certain identifiers (model number, etc.). When Apple releases a new one, the old version of iTunes would need an update to recognize it. (Sometimes this even happens with new device OS/firmware versions.) They bundle these updates with feature and bugfix updates, so you're stuck with a new version.