Do you want to be a teacher? is correct. Want to be a teacher? - is VERY informal, and casual. I never talk like that.

In "want to be a teacher?", the subject of the sentence is implied. "You" is understood to be the missing subject. This kind of construction is fine in informal conversation or a piece of non-academic writing, but, as others in this thread have stated, it is ungrammatical.

This is not proper grammar. Often in informal, spoken language you will hear people drop the "do" as in even though it is not technically correct. The question should be, "Do you want to be my teacher?"