"Not doing" in this sense is to mean the absence of any form of action that originates from thought (for instance physical form like speech, bodily movements) in such a way that reduces the physical forms impact upon sentient beings in its environment.
I realise that I'm heading into one of the unconjecturables here but for practical purposes. I don't think it'll lead me to vexation. This curiosity has arisen through partial insight into karmic laws in which the knowledge of karma became demystified into a rather obvious, almost mechanistic natural working of things. I wanted to try to understand at least from the periphery of the insight, from what Buddha said and from what others practising Buddhism understand so that I can cross-reference the three.
In an example from the Buddha, he chose not to be a father to his son, Rahula (not doing) in favour of seeking his own liberation. His thoughts were of himself. I imagine this would have caused considerable discomfort in Rahula and this discomfort would have been further compounded when Rahula learned of the meaning to his name: fetter or ball and chain.
In an example from my own experience, a person became quite angry with me last week. I remained calm and non-responsive, guarding the senses - essentially watching after myself. I still gave a portion of my attention to the person via eye contact. Because of my "not doing" they became more frustrated. I recall thinking, "this person is lost in their emotions". I received a message from them later that day saying they had cried and let it go.
*As I write this addition, Andrei's answer below seems to resonate here.