What I'm driving at is that the minimal level of interest in politics that it can be rational to have nevertheless invites non-trivial questions. Specifically, if the only belief about politics that you ever bother forming is that no further interest in politics is desirable or necessary, this belief still requires non-trivial justification, and it's irrational to ignore this question. (Which would be implied by the statement that one ignores politics completely.)
The concrete examples of situations where lack of knowledge of politics is costly (i.e. more costly than the cost of acquiring it) are easy to come by. It's enough to observe any occasion of damaging political instability, and how different people end up better off than others because they weren't caught by surprise. There are many other examples too, which I'm sure you can think of.
Now of course, you may conclude that the probability of all this is small enough that it's not worth your time and effort to think about it, just like e.g. the probability of dying in an earthquake is too small to justify obsessing over seismology. But if you only knew that there exists such a thing as earthquakes and that they strike with different frequencies in different places, without knowing anything more about them, it would be irrational not to inform yourself further about it to ensure that the probabilities of these dangers are indeed low enough. Yet while it's easy to obtain this seismological information with full reliability, my belief is that obtaining similarly reliable political information for analogous purposes is much more difficult, and worthy of asking some non-trivial questions.
In what cases you shouldn't just rely on your gut feeling telling that everything is likely fine for the near future, and instead have to work on understanding the situation better? That gut feeling doesn't require additional work, and it does inform you about the current situation.
I'd expect that if anything so serious as to require action on my part was going on, and it would be possible to know it given more effort, my attention would be drawn to it, without the need to research things in advance. (This is the kind of state that I intended when asking for examples, and it's not clear what such examples are.)
There is a tendency to downvote articles and commentaries with a political subtext with a remark on how politics is the mind-killer. I completely understand that nobody wants his mind to be killed, however, I disagree on the employed methods. I don't think anybody can really afford to ignore politics. It's a fact about any group of even a handful of people. Thus instead of shunning politics I think it's better to build one's rational defenses. Understanding that politics is a problem is only the first step. If you stop there, there will always be a big part of life where you are not rational. Therefore I suggest that, as long as it doesn't get out of hands, there should always be room for political discussions if not on the main site at least in the discussion section.