[susan53]

First of all she needs to understand the difference between the two forms - ie need in the negative can be used either as an auxiliary verb (needn't do) or as a main verb - which means that in the negative it needs the auxiliary do and is followed by to - don't need to)

Given that, it's clear that the auxiliary form can't be followed by a noun - modal auxiliary verbs can only be followed by other infinitive verbs. So, if you're talking about cheese for example, you have to use it as a main verb :
We don't need any cheese.
you can't say
*We needn't any cheese.

Similarlarly, when need is followed by an -ing form with passive meaning, the negative can only be used as a main verb. And for the same reason - modal auxiliaries can only be followed by infinitives so the gerund is impossible. Thus :
It doesn't need doing - is OK but *It needn't doing is not possible

However, when need in the negative is followed by an infinitive then both forms are equivalent in meaning and register :
We don't need to do it = We needn't do it. No difference at all - and if that is what was being taught, you're right and the other teacher was wrong.

Even in the past it's the same : we didn't need to do it = we needn't have done it. Both could be followed by either So we didn't or But we did.

Don't know if that helps. It's difficult without knowing the exact examples the teacher was talking about. Any idea?