I have fluorescent lighting fixture under cabinet in kitchen containing two bulbs. Only one works when switched on, except when I reset GFI and remove plug from coffee maker on the same GFI outlet. GFI tests okay. If GFI is faulty, why does is still work to light one of the bulbs only and also coffee maker?

This has nothing to do with the GFI. A GFI either works or doesn't.
Something is wrong with the fixture. The GFI situation is merely coincidental.

Does the fixture have a grounded plug?
How old is it?

The fixture in question is brand new - just replaced because I had the same problem with one of the bulbs not working. Fixture has a ground wire hooked up. It's confusing because both bulbs light only when I unplugged coffee maker and reset GFI. What else could it be.

Have you replaced the lamps. Good grounds are necessary for some lamos to work properly. Also, be sure they are the right bulbs and replace in pairs in a dual fixture.

I have twice put in brand new bulbs. Fixture has a ground wire built in. The bulb that only works when everything else is unplugged from GFI wall socket is good because it does light. It is a 20 amp GFI and kitchen circuit is also 20 amp. Could it be something in the wall switch? Other under cabinet lights that I have around the kitchen also work off this switch and they are fine. Confused?

Are you saying that if you unplug the coffee maker and reset the GFI that both bulbs of the fluorescent lighting fixture work, then if pulg the coffee maker in only one of the bulbs works?

Is the coffee maker on and heating or is it just plugged in?

Have you tried unplugging the coffee maker and plugging something else in to see if both bulbs work?

Have you tried plugging your coffee maker in to another GFI?

If you can plug in another appliance and the bulbs work, I suggest that your coffee maker may have a fault and the GFI is trying to trip but can't for some reason.

Check the ballast in the lamp and make sure the lamps are the right type for the ballast.

There have been hard-start issues with the newer GREEN bulbs in older fixtures. Consider replacing the ballast or at least temporarily swap it out with one of your other fixtures if they are the same.

How do you fix a fault in a coffee maker, if in fact there is one? I need to try and run coffee maker is another outlet (I don't have another GFI in the kitchen though) in the morning when I make coffee and see what happens. I just unplugged the coffee maker, hit the test button on the GFI and then reset, plugged the coffee maker back in but didn't run it, then turned on lights and they both came on.

Your not alone. Fluorescent fixtures on a GFI circuit [Archive] - Mike Holt's Forum

Maybe the solution is a non-electronic ballast in the fixture if it is an electronic one.