How and when does one not invest in others in Buddhism?

These are all quite good answers to my mind. The essential point is that we must always work first and foremost on ourselves, starting wherever we are. The more we clean up our own hearts and minds, the more helpful we will be to ourselves and others- with absolutely no need to assert any doctrine. I wanted to add that at least in Indo-Tibetan Buddhism there's a strong tradition of the teacher carefully examining the student and vice versa before the process of teaching ever begins. So it is really important to consider the situation and to have a student who really wants to learn and enact something you are confident you actually understand before ever presuming to teach. Without that, much better to treat your friends, family and colleagues as quite possibly having more to teach you than you have to teach them.

I have only a little to add as the answer submitted by Jayarava Attwood is quite comprehensive. But here is the little bit I would add.

A couple of things in your question, such as the reference to depending on people to complete their responsibilities, made me wonder if you were referring to a work situation. If that is the case, it's extremely important to not try to impart your religious values on others with whom you work or supervise. It's not fair and one needs to imagine if the shoe were on the other foot and someone was trying to impart their own religious teachings on you. It could be quite unwelcome and depending on where you live, even against various laws governing the workplace.

Modeling the behavior you want others to emulate is always safest. A steady meditation practice can help greatly in reducing or even eliminating the irritation one feels at the behavior of others. It's possible to develop a bit of equanimity within oneself that allows one to function effectively in the workplace without becoming ruffled by those who are less than ideal in their behavior.

And if it wasn't the workplace you were referring to, hopefully something here is helpful anyway. Best wishes. :)

These kinds of questions will always be difficult to answer because we only have one side of the story. We lack the vital perspective that comes from a. knowing you, and b. knowing the people that you are in conflict with. The fact that you are in conflict, but don't seem to be able to acknowledge your part in creating the conflict tells me that I don't have the full story. However, one of my teachers gave me some excellent general advice on communication some years ago: he said that if a communication goes badly, then assume that you have not expressed enough mettā.

In order to relate empathetically one must drop one's own agenda (as teacher or benefactor) and try to feel what they feel and then ascertain the reasons they feel that way. One has to put oneself in their shoes. To do this one must drop all pretence of being superior and of having something to teach them. Drop all the conceit. If you have something they value, they'll ask you about it. That the situation has devolved to the extent you intimate, suggests that you are in conflict with their values. Perhaps you present yourself to them incongruously - you see yourself as having something to teach them, but they don't think of you as having anything to teach them.

I think one is already in trouble if one is modelling oneself on the Buddha or if one is relating to people on the basis of "teaching" them or having something to "impart". It sounds like this is getting in the way of you establishing an empathetic relationship with them. Is the problem that they refuse to see you as a "teacher"? Is that why you are labelling their response to you as "passive aggressive"?

The general principle of Buddhist ethics is that when things go wrong, that we looks for our own unskilful motivations. The unenlightened (dare I presume you are one of us?) are always motivated by attraction and aversion. Sometimes when things go wrong, one must just own up to one's own unskilful motivations. It can be difficult, because it is humiliating or humbling (depending on how you see it). In relating to other people we only have control of our side of the relationship. If things don't go right, we can only ask ourselves if our own attraction or aversion has soured things.

Of course there are times when it is better to walk away. If one realises that one is too entrenched in attraction and aversion to make a change, for example. Sometimes the necessary change is too great to manage in the short term. But one cannot blame the other. For this. If relations break down, it is our own failure. Generally speaking if we are expressing mettā rather than conceit, there is no need for relations to break down. Of course, it's a big ask.

But if the thought is "if only these people would follow my advice" then I would see this as problem of conceit, not of unreceptiveness.