No doubt there are many possible ends an online conversation might serve. For example:
(1) Collaborative inquiry, in pursuit of knowledge and understanding.
(2) Asymmetric teaching, or the imparting of knowledge and understanding to others.
(3) Non-epistemic goals, e.g. creating/reinforcing social ties, "winning" an intellectual competition, etc.
Let's just consider epistemic goals. Discussions between epistemic peers (i.e. #1) tend to be the most interesting and rewarding. It's really wonderful to have the opportunity to improve one's understanding, as when others discern potential flaws in one's initial views. But what about the other case? There are plenty of people out there who are not epistemic peers. Arrogant though this may sound, sometimes you can tell that your disputant is in the grip of a certain confusion, or that they don't really understand the issue, so that there's little you can learn from them.*
The main motivation for continuing, then, will be that niggling itch we feel when "someone is wrong on the internet". They're in a bad state, and so we feel some pull to help make this clear to them, to set them right. Moreover, the act of teaching always has some benefit to the teacher's own understanding, as per my footnote below. So it's not a total waste of time. But if this is time that could have been spent engaging with more insightful interlocutors, this opportunity cost may still be sufficient to deter one whose primary goal is to improve their (own) understanding (or even that of the broader community, if the mistake in question is not widespread).
This relates to my old post, 'engaging persons or ideas?' If we're just interested in ideas for their own sake, then many discussions (i.e. with people who lack sufficient understanding of the relevant ideas) will probably not be worth having. If we think the discussions are worthwhile nonetheless, it must be because we value engaging with the other person.
* = But compare R. Porter:
Almost every new way I have to devise to explain a concept (and sometimes these are pretty simple concepts and thick-headed pupils) makes that concept clearer to me. It's not that the students know something useful that I don't; it's that I need to learn a new facet or explore an uncharted avenue in order to teach them.