Is it safe to use an HVAC system even though the duct wrap contains asbestos?
If you don't trust the expert brought in by the landlord, then you might want to pay to have your own expert validate your concerns.  

As far as to what the landlord does and does not have to do, you'll have to review the lease and laws in your state.  At this point, he has paid for a report of a presumed expert that says the ducts are safe as is; it will be difficult to challenge this without an alternative expert opinion.