In the state of NC does the 3rd brake have to work in order to pass inspection?

Yes, according to this question and answer.

Welcome to Fluther!

Welcome, Yes according to NCDOT:
19A NCAC 03D .0533 LIGHTS. . .
”(c) Stoplights shall conform to the requirements of G.S. 20–129(g).
A stoplight shall not be approved if:
(1) The lens is cracked, discolored or of a color other than red or amber. Minor cracks on lenses shall not lead to disapproval unless water is likely to short out the bulb.

(2) It does not come on when pressure is applied to foot brake.

(3) It is not securely mounted so as to project a light to the rear.

All lights, if present, must be operational. I assume you mean rear brake lights (bulbs) not the dash light that comes on when you use the parking brake. (Your question confused me because I have never heard of anything called a “third brake light.” There are front brakes, 1, rear brakes, 2, and 3rd brakes?? Must be a parking brake???)

@kritiper V V V V
Third brake light on spoiler of Mercedes-Benz

@Tropical_Willie Oops. I was reminded of old Chevys and Mustangs with three tail lights…
Yes, ALL lights must work. (Why lights aren’t checked as part of a vehicle service is beyond me! Lots of money waiting to be made there!)

@kritiper Safety checks in my require all lights be checked and all must be working (check windshield wipers too). I’m in the state the OP is asking about.

No windshield wiper if there is no windshield, I know someone with a Hot Rod took the windshield off for inspection, used a set of goggles.

@Tropical_Willie That’s during a safety check, not a regular lube, oil, and filter service.

@kritiper The OP is asking about the annual

“SAFETY CHECK”

to get your renewal for the vehicle.

@Tropical_Willie I know that. If the lights were checked/repaired/replaced during a service, which they should be, the light would most likely have been working for the safety check and the vehicle would not have failed. Just sayin’ ...