I'm applying for the position of an English teacher - no. You are clearly applying for a specific position (the position) so it is simply described as "the position of English teacher".
There is a difference when you use "position of":
I saw your advert for an English teacher.
I saw your advert for the position of English teacher.
It may help to imagine that "English Teacher" is the job title, so when you use "position of", it is followed by the actual job title.
With a more unspecific position, you may need the article:
Is there a position open for an English teacher?
Do you have any positions for an English teacher?
Student or Learner