This is an interesting topic that has come up a few times at essential oil classes and related conversations. I've heard it stated by a few of the more knowledgeable people that you can't be allergic to essential oils. On the other side of the equation, I know at least one mother with a son that's basically allergic to everything -- especially foods he doesn't like! -- who claims her son is allergic to Frankincense. So who is right? There's a simple answer, but there's more to it than just giving the answer. Let's start with the basics.
Essential oils are distilled versions of various plants, typically plants that are high in molecular compounds that are known to be beneficial. The process basically removes any unwanted aspects (like, say, allergens) and ideally you're left with all the good and none of the bad. Of course, that's only if the essential oils is a high quality therapeutic grade oil -- oils used for scents and in many foods are bought in large quantities and are not therapeutic grade. In fact, most flavorings and scents don't use essential oils but instead use synthetic chemicals that approximate an oil, since it's far cheaper to go that route.
Now, going back to allergens, they're composed of proteins and/or polypeptides, which are large and complex molecules. These molecules simply don't exist in essential oils, because of the way they're made (distilled, pressed, etc.) So when someone claims they're allergic to an essential oil, there are generally two possibilities: 1) they're not using a pure oil and the allergic reaction is being caused by something else in the product, or 2) they're having an allergy-like reaction that's caused either by sensitivity or the detox response of the body.
The first option is pretty straightforward. Inexpensive essential oils can be marketed as "pure" even when they're not -- the AFNOR/ISO standards (Association Francaise de Normalization) is a French government-approved body that sets standards for what should be present in various substances, including essential oils. The problem is, these are minimum standards, and there are many things that are not included as part of the standards, like testing for pesticides and other contaminants. So, a cheaply manufactured essential oil could have all the necessary ingredients to meet the AFNOR standard, and yet it might have allergens, pesticides, or other unwanted chemicals as well. Oops!
This, incidentally, is why some companies have created their own standards, which includes seven tests run on every batch of oils by an independent third party to make sure things like herbicides and allergens aren't present. So, if you've ever wondered why the price of Frankincense can range from $3 to $100 for a 15ml bottle, that's why. Is there anyone that really thinks a $3 bottle of whatever is going to be the same quality as a $100 bottle? Which is not to say it's 33X better, but you get the point. Does anyone think clothing and shoes purchased at Walmart for $10-$20 will always be as good as similar items that cost four times as much? (Don't get me started on Walmart shoes -- I tried a pair once for a couple months before throwing them in the trash! And ironically, it was the most expensive shoe I could find at Walmart, with Dr. Scholl's insoles; they still sucked and now I stick with Nike and new Balance.)
The second possibility is that rather than an allergic reaction, you're instead sensitized to an oil. This can be caused by a detox response (e.g. Grapefruit and Lemon are cleansing oils among other properties, so they can cause itching and other discomfort, particularly if you have a lot of toxins in your body). It can also be caused simply by the strength of an oil -- with a hot oil like Oregano, I'm pretty sure that everyone is sensitive to it! Cinnamon, Clove, and even Peppermint can also qualify as hot oils that many people would find uncomfortable to place directly on the skin. (Peppermint is sort of the odd man out -- it's a "cooling" oil, but at the same time the sensation is quite intense and can feel a bit too much at times.)
That said, sensitivity to an oil can actually end up creating a similar response to allergies. The net result is perhaps the same (or close enough to it), but the catch is that if you're truly allergic to some particular plant (e.g. lavender), it doesn't mean that you'll have the same reaction to Lavender essential oil. You might...or you might not. I've known people that state they're highly allergic to lavender (or some other plant), but the essential oil from that plant doesn't bother them at all. I've also heard stories (though I've never personally met anyone) where the allergic reaction is basically still in full force. If you're willing to give essential oils a try and you have allergies, do so cautiously -- it's your body and you'll know if you're having a bad reaction or not. Start very, very small and increase the amount of oil if you don't experience problems.
Anyway, that's basically it: you're not actually allergic to essential oils, but some of them may not be pleasant to use undiluted and can cause sensitivity, and for some being highly sensitized is just as bad as being allergic. On the other hand, if you're buying lower grade oils you could very well be allergic to some of the other ingredients. If you want top quality, you're going to have to pay more; there's a noticeable difference between a high quality US-based "therapeutic grade" oil and the bulk priced stuff made in China, just like there's a difference between imitation vanilla and an expensive vanilla powder or pure vanilla extract.
Additional reading: The Chemistry of Essential Oils Made Simple, with a pertinent quote: "There has never been a documented instance of an anti-body response to an essential oil. Essential oil antibodies have never been found or detected in anyone. Unless sensitization occurs and antibodies are produced and stored in the body, there can be no allergic reaction. Therefore, we can state unequivocally that essential oils are not and cannot be allergens. Sometimes people do have allergy-like reactions but these are not allergenic in nature. They are detox reactions."