Very rarely people can feel their heart race or feel a bit faint which passes in a few minutes. Lignocaine with adrenaline is frequently used by dentists, so if you have not had a reaction when a having any tooth treatments then you are unlikely to have any issues.
Anaesthetic injections are also available without adrenaline and are used for surgery on the fingers and toes as well as for anyone who is sensitive to adrenaline. Most anaesthetics work within 1-2 minutes and last for 1-3 hours.
It is always important to have a thorough medical consultation and any local anaesthetic injected in a medical setting.