There is through - as in a sword through the heart - and as in blood passes through the heart. 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. This can be more expensive to properly repair (some thousands of dollars.) 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.