Can you use the same chimney for a fuel oil furnace and a woodburner?

I have only one chimney for my fuel oil furnace and my wood burner. The inspector said is was okay but the insurance company took pictures and said I must have a new chimney installed within three months. It is a new insurance company, I dropped my old one and signed up with a new one and they came out and told me I need a new chimney for the wood burner. Anyone know the codes on this and shouldn't my inspector have caught this if it was a code error?