I am sharing this post because I searched and searched to find the fix for this problem and never found the answer. I first noticed the problem when, during daylight driving, the dash lights would dim and the dimmer switch would become active. Eventually I realized that the running lights -- tail lights and headlights -- were also turning on when this happened. No big deal, right? So I let it slide. But the next stage *was* a problem: that turning on the headlights, for night time driving, would not turn on the tail lights. Brake lights, turn signals, and 4-way flasher worked, but no running lights at the rear. Also, the dash lights dimmer switch had no effect.
The fix was (a) install new tail light bulbs, and (b) switch the tail light relay with the inverter relay. It is possible that switching the relays alone would have fixed the problem, but I had already swapped in the new bulbs, so I am leaving them in place. My theory is that the relay grew weak over time, while the resistance of the light bulbs grew stronger, such that the relay could no longer send enough amps to the bulbs to overcome their resistance. It's just a theory, and whatever the explanation, this two-step repair fixed the problem. Of course you could go to the Pick-n-Pull or O'Reilly's and get a new relay. But for $9 out the door for the new bulbs alone, this is a cheap fix.
These relays are located in the glove box area. First you need to drop the glove box, or remove it from its opening. There is a nub on each side that keeps the glove box from falling out of the dash. You push in on each side of the glove box, enough for the nub to clear the opening. You might also find it helpful to pop the telescoping strut loose from its pivot inside the dash. If you do this, the glove box will definitely be falling out of the dash. The relays are plugged into a panel at the top front of the dash opening. Here's a video on where to look: https://www.youtube.com/watch?v=QwujWnQwhck . The video guy says the tail light relay is the third one in the row of three black relays. This is wrong, at least with respect to my '03 Vibe. The tail light relay is the middle one. Although you can't see the row of relays, you can easily reach in and identify them and swap the middle one for the one on the end. Just grab them and wiggle them until they come loose. To reinstall, have the side of the relays with the part numbers printed on them facing toward you.
**EDIT: Please take a look at my second post, below**