As a community gardener who has worked on asbestos litigation cases as an evironmental paralegal, I would stongly suggest that you contact your state's Environmental Protection Agency for regulations and guidelines referencing the proper use of asbestos saturated tar paper and roof tiling materials. There are also specific regulations dealing with the burning of asbestos near agricultural areas, etc.
I would also suggest that you contact your local agricultural agent with your concerns.
While the use of asbestos materials has been discontinued, asbestos abatement projects will continue for decades. It is likely that if the tarpaper is asbestos treated, it is old and needs to be replaced.
The issue with asbestos is the damage it can do when inhaled. Often when it contained by a non, friable layer of mateial, it remains harmless - however, it's removal is considered both prudent and in many cases is legally mandated.