All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The dentist whose work is limited to pulling teeth out based on an orthodontist`s note is similar to a tool in the hand of the latter. Thus, that dentist isn`t responsible for any mistakes made by the orthodontist as regards identifying the teeth to be pulled out. The orthodontist is accused of negligence and misconduct, and so he is liable for satisfaction guarantee.
This case is similar to that of the executioner who executes the order of the judge or ruler in qisas (retaliatory punishment). If the executioner wasn`t aware of the latter`s injustice or mistake and executed the sentence then he isn`t liable for anything; rather, the ruler is liable for it. It is stated in Moghni Al-Mohtaj, "If the executioner was ignorant of the ruler`s injustice and mistake, the ruler is liable for qisas and Diyah (Blood money). Whereas, the executioner is liable for nothing because he is a tool in the hand of the ruler."
However, if the dentist knew that he was pulling the wrong teeth and pulled them then he is liable for satisfaction guarantee. It is stated in Tohfat Al-Mohtaaj: "If the executioner knew that the ruler was mistaken or unjust and still executed the defendant in compliance with his (Ruler`s) order then he (Executioner), alone, is liable for qisas and Diyah."
As evident from the above question, the orthodontist had clearly identified the teeth to be pulled, and the dentist realized the mistake after pulling those teeth. Therefore, the dentist isn`t liable for the mistake made by the orthodontist. It is your duty (Dentist) to inform that patient, but if you couldn`t reach her then you aren`t considered sinful.
Al-Shafie (May Allah have mercy on him) says: "If the doctor knew that he made a mistake which isn`t usually made by his peers, then he is liable for satisfaction guarantee." {Al-Um, V.6:185}.
Al-Khattabi (May Allah have mercy on him) said: "Jurists are agreed that the doctor whose negligence results in any damage or harm to the patient is liable for satisfaction guarantee." {M`alim Al-Sunan, V.4:39}.
It is stated in [Hashiyat Al-Jamal]: "If a doctor treats a patient after asking his permission, then he isn`t liable in case he didn`t make a mistake, but if he did then he is liable for satisfaction guarantee." And Allah the Almighty knows best.