I guess technically all allergies are possible. But dandruff is just dead skin, so you'd also have to be allergic to yourself...
The dust from the dandruff could make you cough or sneeze, or affect you if you have asthma though. But not because of the composition, just of the size/shape/structure.
There is a girl who is allergic to water, which means she is actually allergic to herself (her tears, sweat, saliva...) but I find it highly unlikely you are.