No. Even though the treatment solution for allergy shot is
sometimes called "allergy serum", there is no serum in these
solutions. Serum sickness occurs when a large dose of an animal
serum protein or a large dose of some medications are given and the
immune system make high doses of immune antibody against that
protein. Small capillaries in the joints, kidneys, skin, etc get
plugged up by the clumps formed when the antibody joins with the
protein. Allergy shots use very week doses because the person is
already allergic to the treatment solution. Large doses are
required to develop serum sickness.