What medical conditions prevent someone from receiving allergy injections?

None. Some persons, such as those who are on high blood pressure medications called beta-blockers, may be at increased risk for developing serious allergic reactions from allergy shot injections however; the decision to begin anyone on allergy shots begins with the careful analysis of risk vs. Benefit.
Allergy Shots. Typically allergy shots are not given to patients on beta-blockers, sometimes on ace-inhibitors, and oftentimes when an immunodeficiency is present.....Sometimes this includes patients with HIV (although HIV patients on appropriate arv therapy can receive allergy shots) but one would need to discuss with their doctor.