In real life, an anesthetic is used to stop you from feeling physical pain. If you receive an anesthetic in a dream, you want to numb yourself from emotional pain. You may have recently had a traumatic experience and your dreaming mind knows that you are not yet ready to deal with it. Your dreaming mind is trying to protect you from further trauma.
A dream about being anesthetized can also be a message that you are trying to do too much. Your dreaming mind could be telling you to sit back and let other people take care of things for a while.
Dreaming that you have been given an anesthetic can mean that you are afraid of losing control.
If you dream that someone else has been given anesthesia, you feel that they are putting too much pressure on you in waking life, and you want them to calm down and be quiet.