I was supposed to answer the question "Did you post my letter?" I answered "No. I have forgotten to." It was a wrong answer. The right one was "No I forgot to."
Can I use Present Perfect there? On one hand since now I remember that I was supposed to post the letter, absent of this knowledge does not prolong up to now. But on the other hand, there is a consequence of my forgetfulness that is relevant now—the letter is not posted.
I don't know who told OP that "No. I have forgotten to" was a "wrong answer", but it was bad advice.
Since the "forgetting" is still "relevant to the current moment", Present Perfect is grammatically/logically acceptable in such contexts.
But it is true that in practice most native speakers would use Simple Past "No. I forgot to", because very often the actual action described using Present Perfect continues into the present (i.e. - when you say "I have lived there all my life", it always means you still live there).
The tricky thing about forgetting to do something is that effectively it means you forgot about it. But as soon as you admit to having forgotten about it later, you're no longer forgetting. Because it creates a bit of semantic/temporal confusion, natives speakers tend to avoid the implications of Present Perfect here.
