Whether you are traveling or live in Tokyo, you may need to find a dentist on Sundays or public holidays. There are few clinics which are open on Sundays and public holidays, but sometimes doctor does not speak English enough to make you feel comfortable to have treatment. In Tokyo, most wards do operate Dental emergency clinic on Sundays and Public holidays. However attending dentist takes turn from the dental society, so the clinic may not be equipped to provide treatment like root canal treatment or esthetic temporaries, and often only medication are giving for tooth nerve pain. I am aware that Minato-ku, Shibuya-ku, Shinjuku-ku dentist takes turn to provide dental emergency care on Sundays and public holidays at their own clinic. This means they can provide dental care with their own set up just like their office hour with their own staff and hence provide more services. Please call the clinic and find out if the dentist speaks English before you go and what kind of problem you are having. If you can take a picture of your tooth with your phone and send it by mail beforehand, dentist can better assesses your situation.
I have listed link to the dental offices which works on specific public holiday. Unfortunately dental clinics changes with date and are written in Japanese. Please have some one help you read the webpage if necessary. You can try to figure out the dates and phone number if no one is around to help you.
List of attending dental office for Holidays in
Shibuya Ward http://www.sibusi.or.jp/dayoff.html
Shinjuku Ward https://shin-shi.or.jp/guide/sos/