Learn English – As when-does it make sense

Both "I won't be needing that" and "I don't need that" are grammatical.

The problem with your teacher's logic is that "need" is not a modal in either of those. Two easy-to-detect signs of modality are:

• They have the syntactic properties associated with auxiliary verbs in English, principally that they can undergo subject–auxiliary inversion (in questions, for example) and can be negated by the appending of not after the verb.
• They do not inflect (in the modern language) except insofar as some of them come in present–past (present–preterite) pairs. They do not add the ending -(e)s in the third-person singular (the present-tense modals therefore follow the preterite-present paradigm).
  _Wikipedia

For the first point I listed, it should be pretty obvious that not does not follow need in "I don't need". For the second point, since people say "he needs...", we know it's not modal in this sense either.

"Need" specifically is a semi-modal. Your sentence needs modification were it to use need as a modal:

I need not have help.

According to the statistics here, the modal form of need is not used very often in American English (which is why I think it sounds old fashioned). A COCA search for BE needing (capitalizing "be" means it matches all forms of the verb, such as be, were, 're, etc.) returns 356 results, so I think it's safe to say that educated speakers do use "will not be needing" and similar (I would).

You are right. But means only in this context. And this use of but makes the sentence formal. Here are a few examples:

We can but guess at the extent of the problem. = we can only guess...

Several Japanese companies already operate in the UK – Nissan, Honda, Fujitsu, to name but a few. = ... to name just a few

Lots of interesting different flavours combine - mixed spice and wild berries to name but two.