What are the legal effects of an inoperative anti-collision light?
The answer to your question is in FAR 91.205:

91.205(b) - Visual-flight rules (day):


  (11) For small civil airplanes certificated after March 11, 1996, in accordance with part 23 of this chapter, an approved aviation red or aviation white anticollision light system. In the event of failure of any light of the anticollision light system, operation of the aircraft may continue to a location where repairs or replacement can be made.


and 91.205(c) - Visual-flight rules (night):


  (3) An approved aviation red or aviation white anticollision light system on all U.S.-registered civil aircraft.  (it goes on to talk about color, which I've cut out) In the event of failure of any light of the anticollision light system, operations with the aircraft may be continued to a stop where repairs or replacement can be made.


So as you surmised in your question, a 1967 Cherokee is required to have an operating anti-collision light for Night-VFR operations (because the night anti-collision requirement is retroactive and doesn't care about certification date), but not for day Day-VFR operations (because it was certificated prior to 1996).



If you HAVE an anti-collision light it is required to be operating subject to FAR 91.213. The relevant part of that regulation says that you can take off with inoperative equipment provided that equipment is not:


  (i) Part of the VFR-day type certification instruments and equipment prescribed in the applicable airworthiness regulations under which the aircraft was type certificated  


You're OK for Day VFR because CAR3 (and FAR 23 prior to 1996) didn't require an anti-collision light for daytime operations.  


  (ii) Indicated as required on the aircraft's equipment list, or on the Kinds of Operations Equipment List for the kind of flight operation being conducted;


Your Cherokee predates the concept of a Kinds of Operations Equipment List in small aircraft by several decades, so your reference is 91.205.


  (iii) Required by §91.205 or any other rule of this part for the specific kind of flight operation being conducted; 


If the kind of flight is "Day VFR" the anti-collision light is not required by 91.205, but if you want to fly at night it is required (as we determined previously from 91.205).


  (iv) Required to be operational by an airworthiness directive


No such Airworthiness Directive exists for anti-collision lights on the PA-28 family.

…so per 91.213 you can placard the beacon switch INOP and "deactivate" the anti-collision light (simply putting a bit of masking tape on the switch holding it in the off position &amp; writing INOP on that is generally considered adequate) and fly in Day-VFR conditions (ideally to an airport where you can get a replacement lamp for your beacon).

Note that the fact that your beacon was optional equipment doesn't enter into the decision tree for 91.213: Any installed equipment gets evaluated according to this regulation. As an example my (1965) Cherokee has both a tail beacon (originally installed by Piper) and a belly strobe (added by someone over the years): if either is inoperative it needs to be "deactivated and placarded" appropriately.