Elie was able to keep his gold tooth by a combination of wit, compassion, and luck.
When Elie realized what was happening (that gold teeth were being extracted), he thought of a plan. He feigned sickness. The dentist let him go and told him to come back in a week. In a week, Elie returned and gave the same excuse.
The dentist allowed him to go again, probably as an act of mercy. Elie mentions that the dentist was a fellow Jew from Czechoslovakia. Elie is not sure, but this could have played a role in the leniency.
Shortly after this event, the dentist's office was shut down and the dentist was sentenced to death. It seems that the dentist was making a profit off the gold.
Here is a quote, which sums up these three points:
"All right, son. Come back to see me when you feel better. But don't wait for me to call you!"
I went back to see him a week later. With the same excuse: I still was not feeling better. He did not seem surprised, and I don't know whether he believed me. Yet he most likely was pleased that I had come back on my own, as I had promised. He granted me a further delay.
A few days after my visit, the dentist's office was shut down. He had been thrown into prison and was about to be hanged. It appeared that he had been dealing in the prisoners' gold teeth for his own benefit.