"Red wine headache" syndrome, as it's often called, affects a surprising number of people. But the culprit isn't sulfites, contrary to popular belief. Sulfite allergies, which are relatively uncommon, can result in wheezing, dizziness, accelerated heartbeat, hives, and, in very rare cases, both of one's arms falling off abruptly, crawling by the fingers across the floor, and trying to throttle one's dog.
OK, the last symptom was a lie. But one thing that is true about sulfites is that your average package of dried apricots contains radically more sulfites than a bottle of wine (about 2000 ppm vs. a legal maximum of 350 ppm; most wines have less). So if you really want to see if you're sulfite-allergic, eat a few dried apricots and check out what happens. Just make sure your dog is parked safely outside.
Regarding those headaches, though. The likely culprit seems actually to be a group of chemicals known as biogenic amines, which include tyramine, histamine and others (though this is not conclusively proven). Nevertheless, because the juggernaut of science stoppeth for nothing, researchers at UC Berkeley have developed a suitcase-sized gadget that will detect amine levels in a drop of red wine, as described in this interesting article in today's Press Democrat. Haul it with you to a restaurant, and you may just find yourself saying "no thank you" to that bottle of 1961 Château Lafleur after all.
Or you may just want to switch to wine made from cherries. Which apparently, according to Dr. Vasantha Rupasinghe, the Tree Fruit Bio-Product Research Chair at Nova Scotia College, has practically no detectable histamines at all.