Take lost of photos inside and out, document, document, document everything in writing. Demand any costs estimates, statements of work, and invoices in writing and save copies. Write down dates and times and names when you call any of the involved parties.
Without seeing the damage, it's difficult to say what will be required. The furnace installer should bear the cost of this. The vent obviously has to be properly installed, any moisture damage to the roof, walls, ceilings, insulation, etc will have to be repaired. Be very involved in this process, talk to the local building code officials/inspectors about this, be there while the repairs are made if possible, talk to the contractor performing the work, watch what is being done, take before/after photos. You don't necessarily have to be rude or angry, but make sure they know that you are seeing what's going on and that you expect all problems to be completely resolved. It may be worthwhile to you to spend a little to hire someone to thoroughly inspect the attic and surrounding areas for any hidden damage (or if someone you know or are related to is capable of this, so much the better - just a neutral 3rd party who is knowledgeable would be good).
I'm far from an expert on this sort of situation, but I think that's how I would handle it if it were me.