Similar things have been suggested before:
All of these questions have a common theme. Some user or users are asking questions, then not accepting the answer. Sometimes the complainant will note that the community has clearly chosen the best answer with their upvotes. Sometimes the complainant will simply note that the user has done this before and that it is obvious that they are doing it on purpose.
First of all, every post has two measures of success: Upvotes and Checkmark. They are actually completely separate.
Getting the checkmark simply means that the OP decided that the post fulfilled their needs best. No one else gets to decide this one, because it is strictly a measure of "did this post directly help/solve the issue for the asker."
Getting an upvote means that the people who are reading the post, (most of which are not currently experiencing the same technical problem) think that it is good or sound advice. This is where the vast bulk of reputation will be accrued.
How does it affect the individuals or the community when the OP does not accept answers?
It is annoying to feel that you created a perfect answer, you put your time and effort into this question and you did not get the checkmark. As noted above, this doesn't necessarily mean that the OP isn't paying attention, it's possible that your answer wasn't what they were looking for (I.E. "Don't Do It." as an answer to "How Do I...?"). This creates a negative experience for the individual user. The user has a few options for recourse.
- Bring it to Meta. Suggest punishing the user for asking questions.
- Continue as you were, attempting to help people as much as possible, disregarding your own reputation sometimes.
- Ignore that user and any other users who have low acceptance rate.
Option 2 and 3 seem the most reasonable to me, in fact 2 is probably the best because it still allows you to get reputation from the other people viewing the questions.
Additionally, if the OP does not "accept" an answer, then the highest voted answer will remain at the top, so if your answer is the "best" and you feel that you deserve the checkmark, at least it will remain at the top of the list, which means it is more likely to gain upvotes than the other answers.
How to "unaccepted" posts affect the community? They really truly don't. That's a short and glib summary to the much longer answer of "they sort of do."
It could be argued that it presents a problem, in that someone who google/bing searches an issue, cannot be certain that the highest upvoted response necessarily fixed the issue without that checkmark being there. However, people tend not to understand the difference between "highest voted" and "marked as correct", and pedestrian question viewers will often just upvote the checkmarked or highest voted answer, regardless of it's correctness.
So ultimately, it does not affect the community very much, because the information is still there. And that is what StackOverflow is about. It is not about reputation, it is not about the individuals, it is about the sharing of knowledge, and creating one place where all your technical questions can be answered. If you created a solid answer to a good question, then you have benefited the community. If that user created a solid question that was interesting and allowed you and others to create good answers, then that person has also benefited the community.
This person is contributing good questions to the community. I'm not going to punish them for that.