This is pretty obvious, but if you're allergic to bees, you probably shouldn't become a beekeeper.
Also, if you're a child under the age of one, it's recommended you don't eat honey, as it sometimes contains a form of a dormant endospore of botulinum, which can be dangerous—and even fatal to young infants under the age of one.
(Even though most parents know about this danger, it's a good idea—if just to save yourself from lawsuits—to have a warning label on your honey jars.)
You will get stung from time to time; it happens. If you're not terribly allergic (most people are a little allergic—hence the little bump that arises after you’ve been stung, not unlike mosquito bites), you'll survive and eventually even get used to it. Beekeeping is not among the top 20 "most dangerous jobs" list by Forbes. In fact, we doubt it'd make the top 100.