tl;dr: 1. Don't tell your superiors nor HR. 2. Consider staying and being politically active.
Suppose one is happy at one's institution until something happens that makes it impossible to stay given one's strong ideological beliefs.
Are you sure it's impossible to stay - or is it impossible to stay and not get highly active in trying to influence surrounding society? If you've lived someplace long enough, if you've set down some roots, if you feel it's important to you, perhaps you should stay and struggle.
In my public/political activities, especially within academia (mostly union business), I was very exasperated at people who agreed with us just leaving or withdrawing.
For the rest of this answer, suppose that you're leaving and it's final.
Is it a good idea to mention these beliefs if they are the main reason for wanting to move?
Faculty-management relations, and to a great extent even junior-researcher-tenured-senior-researcher relations are too confrontational in economic class terms for you to be committed to be forthcoming with them. Specific circumstances might differ, but - unfortunately - they're not your friends and are not committed to you and yours. Only tell them what they need to know.
As for personal acquaintances at work - now that's different; it depends on your specific relationship with each individual.
But of course - you could also go for the entirely-public option, if the reason you're leaving really pains you: Making a public statement at some faculty event, publishing an open letter to all faculty and local press, putting up posters as you leave etc. That's again if you think it'll have an effect and are willing to burn some bridges.
Now for your examples:
"Why do you want to study at this university?" Because this university is in Russia and I approve of Putin's actions as President.
Bad example, not buying this at all. I mean, if you said this, people would assume, and be right, that you're just giving them a bad lie, or rather, that you're not willing to tell them the truth and are mocking them instead. Now, that's irrespective of whether you approve of Putin's presidency or not; people who do would not move to Russia because of it. So I'm ignoring this example.
one could equally have "I want to leave Russia because I disapprove of Putin's actions as President" etc.
That wouldn't be convincing either. If you were worried about some specific actions of the Putin-led government, that's something else, but this sounds like a lie. Plus, Putin has been in power forever, and you've just decided that bothers you? ...
"You've worked at this institution for 10 years. Why do you want to leave now?" Because it's in the UK and after the Brexit referendum, I no longer feel welcome here.
Yeah, so this is exactly what I was talking about. Try: "It has been 10 years - a full decade! I feel I need a change in my life - I'm too settled into my routine." Or tell them it's for personal reasons relating to your family, and say you'd rather not discuss the details. Or whatever works. Better not to lie, either; just list other priorities that weigh in favor of leaving.