Are there any breathing techniques to relieve ...

I don't know of any, but self-hypnosis has worked for me in the past.  It works too well from my experience if the tooth is still in your head as the problem can get worse and you will not notice it as you should.

Does this prohibition cover topical drugs too?  Or is it painkillers in pill and injection form that is the problem?  I'm trying to see the harm in putting lidocaine on a tooth or the gum and I don't see any off the top of my head.  As far as a drug, lidocaine has no effect on a body except in numbing pain receptors where it is physically applied.  It is a member of the same family of drugs as cocaine, procaine, and novacaine.

I hope whoever has the toothache gets well soon. 

I don't know of a breathing technique, but you might want to check out EMDR.

EMDR

I am a dentist, and I know of no breathing techniques that are specific to relief of dental pain.

 That being said, you may benefit from the more general relaxation breathing techniques.  Since you asked this question, I am assuming that you are familiar with them.  If you are not, just let me know and I will post them to the best of my ability.

Mike

First I have to ask you WHY you are attempting to do this.  Are you trying to avoid local anesthetic ("novocaine") ?  If so, you will have to practice these techniques for a LONG time before you will be able to use them for your intended purpose.

Basically, you need to practice deep breathing, clearing your mind, and TOTALLY relaxing your body.  It isn't so much just a breathing technique as it is a form of meditation. 

I am NOT an expert on this, so I am going to recommend that you either do an internet search on meditation or that you contact someone in your local community that will be able to teach you this technique in person.

Good luck !

Mike