Using "the" would be somewhat odd.
In a sentence like "The teacher, Mark, will tell us more", it is understood that there is only one teacher. It could be possible (for example if a police office, social worker and teacher were in a meeting), but it would be rare. (I've adjusted your grammar to will tell and added punctuation, based on what I think you mean)
Using "a" is possible:
In the sentence "A teacher, Mark, will tell us more." The name of the teacher is additional information, it implies that the main fact is that a teacher will tell us something. The name of the teacher is secondary. On the other hand "Mark, a teacher, will tell us more" introduces Mark, who will tell us something. That Mark is a teacher is additional information. In either case a relative clause could be used instead "Mark, who is a teacher, will tell us more"
Use of professions as titles, without any article, varies between dialects of English. It is less common in British English. I would not normally say "Teacher Mark"
Use of "Prime Minister" as a title is considered an error by many British speakers. It is a common Americanism to say "Prime Minister May", British speakers will more often say "The Prime Minister". Some job words can always be used as titles, for example: President, Professor, Officer, Doctor.