Is it permissible or forbidden to correspond with a young male teacher, bearing in mind that she has not seen his face because she wears a niqab?

It is not permissible for a woman to establish any relationship with a non-mahram (definition) man, as this involves following the footsteps of Satan and transgressing the boundaries of Allah. The sister must immediately sever this relationship. The Sharia (definition) came to block the avenues of temptation, and this includes prohibiting women from speaking with men except from behind a veil, using speech that is neither soft nor submissive when necessary. Allah Almighty said: "And when you ask [his women] for anything, ask them from behind a partition," and "then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech."