It's not a relay. It's the stop switch in the headlight motor has bent, corroded, or dirty contacts and/or shaft.
The headlights always have power even without the keys. The motor is designed so that when the headlight bucket raises or lowers it hits a stop. This stop causes the shaft inside the headlight motor to torque up and shift. This little movement hits a set of contacts turning power off for that direction. Either the contact is bent or there is too much grime in the motor reducing how much the shaft shifts.
If the motor is still running, Then years of dirt and grime have built up on the contact or shaft for that direction. To repair it you need to drill out the rivots and open the motor. You will see the contacts at the top end and bottom end of the shaft. Just clean out the motors of all grease/grime, lube with silicone grease then clean the contacts off with some 1000grit sand paper and re-assemble the motors with bolts. If you are planning on opening the motor tor repair it, It would be a good idea to replace the bushings that the gear rides on. Over time these bushing wear down and can cause problems like getting grime on the contacts and shaft.
You can get rebuild kits from the Fiero Store as well as other parts:
http://www.fierostore.com/Product/Browse.aspx?d=182&p=1
FYI on Fiero headlights:
84-86 used one type of motor/system and 87/88 used a different style.
The headlight motors are the same ones used on Fierbirds but reversed. The Fiero left side headlight motor is used on the Firebird right side and visa-versa.
If your gear strips, A quick fix is to open the motor up and flip the gear 180*. The motor only uses half of the gear. So if it strips out, The other half isn't bad.
Did you know that to clean the headlights on a Fiero you can push the parking lights on then lightly put pressure on the headlight side. This will raise the headlights up but not turn them on. Pushing the parking lights off will then lower the headlights.