Correct would be:
"Neither I nor my teacher plays tennis or smokes."
But this sounds very formal, and you wouldn't ordinarily be so formal when you're talking about tennis and smoking.
I'd say:
My teacher doesn't smoke or play tennis, and neither do I.
Incidentally, are you Hispanic? You've used the word "doubt" where Anglos would say "question." It's a usage I've often heard in Latin America, and it's not really good idiomatic English.
regards
edward
I have a serious doubt...is it correct to say: `Neither me nor my techer play tennis or smoke.`?