If a shoulder harness becomes INOP in an aircraft certified before 1978, is it still required equipment?
Shoulder harnesses have been required for all aircraft manufactured since December 12, 1986 for all seats, July 19, 1978 for front seats.
14 CFR 91.107 (2) and (3) is pretty clear:

(2) No pilot may cause to be moved on the surface, take off, or land a U.S.-registered civil aircraft (except a free balloon that incorporates a basket or gondola, or an airship type certificated before November 2, 1987) unless the pilot in command of that aircraft ensures that each person on board has been notified to fasten his or her safety belt and, if installed, his or her shoulder harness.
(3) Except as provided in this paragraph, each person on board a U.S.-registered civil aircraft (except a free balloon that incorporates a basket or gondola or an airship type certificated before November 2, 1987) must occupy an approved seat or berth with a safety belt and, if installed, shoulder harness, properly secured about him or her during movement on the surface, takeoff, and landing.

So what I read above can be interpreted as &quot;if it's installed, it's required&quot;.
Some aircraft required an STC to retrofit the belts, you'll need to find out if that is the case (if it is, you can't remove it, but if there is no STC it could possibly be removed, a call to your local FSDO can clear that up). However I've been told that if the aircraft is equipped with shoulder harnesses, then they are required to be used.