Aside from running the risk of hurting yourself, you can mess up - real bad, first of a lot of local anesthetics usually contain a cocktail of the anesthetic agent and adrenalin(epinephrine), adrenalin is useful because it induces contraction of the blood vesicles at high concentrations that results in more precise anesthetics, you require a lower dose as well as the prolongs the effect. Though if one uses this cocktail at an end-artery fingers, toes, penis etc. you risk cutting the blood flow to that part of your body, this can result in tissue damage even necrosis (cell death). If you by mistake inject a local anesthesia into an artery/vein there can be rather adverse effects as well.