As a nurse, rarely have seen petechiae around a venipuncture site; have seen plenty of ecchymosis (bruising) as well as local reactions (rash) to the adhesive of the tape, bandage, etc. These rashes can resemble petechiae, but they are NOT petechiae.
Petechiae are broken capillaries that appear as red spots. If the red bumps appear in a round area surrounding the puncture, they could be petechiae or a reaction to the metal on the needle (foreign body reaction. It they appear in a band where the bandaid, plaster or tape was, they are most likely an adhesive allergy. Pretty common and either of these should resolve in a few days.
Bob