Hi mesmerred,
A microchip is usually the size of a rice grain. To chip an animal, the chip sits inside a special needle which it then inserted under the skin. The microchip is pushed out and the needle withdrawn.
Animals may find the chipping process more painful than a standard vaccination, simply because of the larger needle size - many puppiers I've done tend to yelp and squirm a bit (and to be honest I can't blame them!). Once the chip is in, however, they shouldn't feel anything.
I would have thought an intramuscular injection would be more painful than one under the skin - the site below states that due to the size of the chip compared to that of the bird an anaesthetic is used:
http://www.penninevets.co.uk/birdchips.php
However once the chip is in, the bird shouldn't feel it, so I don't think it would try to peck it out. That said, I haven't any experience in chipping birds so I'm not entirely sure.
The best thing to do if you are worried is to phone your vet and ask them - they will best be able to advise you.