You want a solid rain proof covering of tar paper. If the tears can be fixed with new paper that has the proper overlaps and underlaps then it should be ok without replacing it all. The tar paper on the roof should shed water without getting the decking wet.
If your decking is wet now then you don't have a good paper covering now but I think you are saying that the paper is wet. Wet paper should dry fast and shouldn't be a big problem installing shingles over it. If the decking just got wet in a few places where the paper is torn, then it should dry but check to make sure the decking didn't get soft, warp, and fall apart easily.
I would be concerned if the paper is buckled enough that would cause bulges and buckles in the shingles. If the paper has big bubbles then they should be cut and covered. If all over, then think about all new paper. Get plenty of pictures. Get your guarantee in writing. The shingle companies won't pay for bad installation. It may be time to get a new company with new paper, etc. It wasn't your fault he waited for two weeks was it? If this is a new house, then maybe you may want to tell him you won't accept and pay for it without new paper and any bad decking replaced. Good luck.