Well, you could ask my parents! (I'm a first-born, and sometimes fit the stereotype a little bit too well . . . )
This might not relate so much to parenting . . . but one thing someone told me recently has helped me personally - the desire for perfection isn't intrinsically wrong (considering that God has given us a desire for heaven and for righteousness). That doesn't mean we can't mess it up - but it helps to realize that the basic desire isn't wrong, it's how I go looking for it that can get me in the wrong place (trying to have that perfection here and now, doing it my way in my strength).
I agree with Jessica. As a fellow first-born perfectionist, I was (and srill struggle with being) driven by fear of man and people pleasing. The perfectionist child sees the smallest mistakes & missteps as the end of the world and a complete and total failure to live up to the impossibly high standards they set for themselves. I would continually remind your child that their self worth comes from nothing they do or don't do - that the Scriptures are our standard, that only in Christ do we have real worth, and that He truly WAS perfect because we can never be.
Point in case, I just edited this post because I realized I left out the word "to" in my 3rd sentence. I tried to let it go - but I couldn't!!!