Hicathy,
That’s a possibility but unlikely. It would depend on the type of allergy cover.
If it was plastic and had no ventilation at all then it could trap moisture inside the mattress with no airflow in a dark, humid, and warm environment and contribute to the possibility of mold or mildew in the bottom area of your mattress typically in the bottom of the mattress where it sits on the base (foundation or platform bed).
Most allergy covers though would be breathable enough to prevent this unless there were other risk factors involved (see post #10 here).
Phoenix