In either case, a gas appliance can not be vented into the same flue as an oil appliance and there is no need to do so. A gas dryer can be vented directly through the wall to the outdoors. It is very inexpensive to do so. The oil furnace requires an excellent high-temperature chimney.
Oil furnaces require a properly-sized liner inside an old tile chimney.
Patching the hole left after removing the gas drier may not be sufficient for fire safety. See about getting the chimney lined by a heating expert.