When you have an allergic reaction to fish, your body is reacting to certain proteins in the fish that it mistakenly believes are harmful. Your body uses antibodies and other chemicals to attack the fish allergens, which is why you suffer allergic symptoms if you eat fish. Although fish oil supplements don't usually have those allergenic proteins in them, people with fish allergies are generally advised to avoid fish oil. There could still be traces of fish protein in the oil depending on how well-processed it is.
Whether or not you have an allergic reaction to fish oil supplements depends largely on how sensitive your allergies are. Some people with fish allergies can't even have fish gelatin. Most commonly, the allergic symptoms displayed by people with fish allergies who take fish oil supplements are skin rashes. There are side effects to fish oil supplements that afflict people without allergies, too -- particularly diarrhea.