I think the plastic is more to keep moisture, dust and dirt from the inside of the door out of the cab. You should be able to get any clear poly from a say Lowes, Home Depot or the like. As for the black stuff, just some silicon type sealant should be fine.
As to your question, I don't think its 100% vital to have the plastic on there, but you will most likely start to notice more dust inside the cab, also the possibility of mold growing on the back of the door panels.... Better to keep on there or replace it with Dynomat or some other type of sound proofing material.