I had to drop by a dental clinic to treat my teeth. They said I didn’t need to pay treating my teeth. Because they guarantee free treatment in 3 years. 听起来自然吗？

"I had to drop by a dental clinic to have my teeth treated." The people at the clinic treat your teeth, you don't.

It is correct, but unnatural to say "treat my teeth". We tend to use "treat" with illnesses or diseases. A doctor treats a headache, and a dentist treats a toothache.

I might say, "...to have my teeth examined."

"They said I didn't need to pay to have my teeth examined, because it hadn't yet been 3 years."

"they guarantee free examinations" sounds strange to me. This is both a problem of language and culture. Our healthcare system doesn't work this way. I'm not sure how to translate this sentence. I would say, "...because it hasn't yet been 3 years since the last time I paid for an examination." I can't think of a more concise way to say this right now.