Long story short. I ended up putting oilfurnace in 5" s.s.flue in 12x12 unlined brick chimney and woodburner in 6"s.s. flue in same cavity as other. looks a little stupid where they come out up top but they work fine.
Technically, that is against code also - there is supposed to be a certain minimum separation, (4" of masonry IIRC)or both flues need to be Class A pipe in order to avoid the risk of a fire in one compromising the other.
As to the OP question, basically you have the answer in the prior posts -
1. The HIGH risk of backpuffing smoke from one appliance into the other, and then into the living space.
2. The Oil burner automatic draft damper causing problems for the draft on the wood stove.
3. There are some concerns about mixing the differing exhaust chemistries - potential for corrosion problems, and / or increased deposits.
4. Not mentioned, but each appliance will have it's own cross section area requirements, and a shared flue would need to be big enough to accomodate both running at the same time - but that would probably exceed the maximum allowed on the wood stove. (In the OP's case, he doesn't have the combined area, but the wood stove is marginally OK)
The rule for many years has been "One Flue / One appliance" - there are only two narrow exceptions that I know of -
1. You can share a flue between a furnace and a hot water heater IF they are both the same fuel (gas or oil) and you meet certain requirements for placement.
2. There are a couple of multi-fuel (oil or gas and wood) boilers that are UL LISTED to share the same flue.
Bear in mind that the codes mostly aren't written to "make installers money" - they are written because people DIED doing it some other way... I had a friend (now deceased) who used to refer to the NEC as "The book of the dead" for that reason.
Gooserider