Local anaesthetic does not break the fast, because it is not food or drink, and is not equivalent to them.
Sheikh Ibn Uthaymeen (may Allaah have mercy on him) was asked: With regard to anaesthetic which is applied to a tooth during the day in Ramadaan " do I have to make up that day if I have this anaesthetic?:
He replied: No, because anaesthesia does not break the fast. A local anaesthetic only affects the place where it is given, it does not reach the stomach. So if a person is given an anaesthetic when he is observing a naafil or obligatory fast, his fast is still valid. End quote from Fatawa Noor ala l-Darb. See also: Fataawa al-Shaykh Ibn Baaz (15/259).
But if it is a general anaesthetic that leads to unconsciousness that lasts all day, then the fast must be made up. Ibn Qudaamah (may Allaah have mercy on him) said: If a person loses consciousness for the entire day and does not wake up for any part of it, then his fast is not valid. This is the view of our imam and of al-Shaafai. Then he said: If the one who lost consciousness wakes up for part of the day, then his fast is valid, whether it was at the beginning or the end of the day. End quote from al-Mughni (3/12).
Based on this, if the fasting person is given an anaesthetic whilst fasting, then his fast is valid and it is not invalidated by that. But if he was given the anaesthetic before Fajr, and he remained asleep under the influence of the anaesthetic until the sun set, then his fast on that day is not valid.