How relevant is "moving on" as an argument in peer review? For me, publishing results only when they are final, complete, and positive is more of a tradition than a necessity. I would find it valuable if others were able to publish what they have tried, even if they have not achieved what they (initially) aimed for. Unfortunately, this is very often not possible, at least in peer-reviewed journals.

I always pity the results that are lost because of this; in particular, when reviewers ask for additional research that is feasible in general, but infeasible because the authors are busy with other projects, moving to a different institution, leaving academia, or whatever other reasons. If I were an author in such a situation, I would probably share this reason with the reviewers to explain why additional work is not possible. As a reviewer, I might consider this in my recommendation. My question is: how relevant is this? Does this happen a lot? How likely is it to influence the reviewers' decision? And will the editor care?

These questions are very hard to answer, so good answers might be based on experience rather than hard facts. I would very much welcome those.
The way I see it is as follows: the reviewer's primary job is to judge whether the main claims made in the paper are borne out by the evaluation. The editor's primary job is to fill the conference/ journal with the papers that had the most interesting claims and passed review.

If you send in a paper, and the reviewer complains that additional research is needed, you actually have two options: you can add the research, or you can reduce the strength of your claim. If the editor complains that the claims are not interesting enough, you should look for a less prestigious venue.

So in short, it's perfectly fine to publish unfinished or negative research, but you need to be very honest about it.