People who are allergic to colophony (which is used in Elastoplast adhesive) will develop contact dermatitis (See Fig. 2 here:
http://www.allergy-clinic.co.uk/documents/medical-articles/Contact-Dermatitis-CACI-2004.pdf ). They should also take care to avoid many other products:
http://www.bedfordhospital.nhs.uk/upload_folder/patient%20information/allergy%20to%20colophony.pdf
I've never heard of it resulting in anaphylaxis (which is the potentially lethal shock that other allergens, e.g. wasp stings or peanuts, can cause in some people). Further, googling seems to find no link between the two.
I've been in charge of first aid in a comprehensive school for many years (and duty first aider at a railway station, and several other places, in later years) and I've always been happy simply to ask the person whether they're allergic to sticking plasters or not. (I'd probably use a couple of dozen plasters most weeks at school and I can't recall killing any of the kids through doing so!)