I know I'm late to the party here, but here is my take.
In the particular scenario you site, the responsibility for the audio not working could fall to a number of people.
1. Tape operator. If he/she pulled down the playback pots while testing, and did not return them to their detente position.
2. Editor. Did the editor edit the correct sound onto the tape? How much time did anyone on the crew have to test?
3. Engineer in Charge. He/she is responsible to all the equipment and represents the equipment rental company on the job site. If the deck is broken, then he is responsible to fix it or replace it. (This is really a mobile TV position and probably doesn't apply.)
4. Audio. Did the mixer track the audio when the director said, "Roll tape, Track tape?" The director did say to both roll and track it, right?
5. Director/producer. Was enough time alotted to rehearse? Would the mistake been averted with a proper rehearsal?
6. Technical director. Only if the technical director was directly involved with what I think of as other job functions. Did the TD set up the equipment? Did the TD direct or confirm that an audio - tape facilities check was performed? Did the TD actually roll the tape? Was the TD the one who edited the tape? Was the TD asked to do a technical element rehearsal?