I don't put my players into no-win situations. Feel free to run your games differently, just don't expect me to stick around if you force the issue without an alternative.
See, the thing is, if you don't remove player agency, it's never a "no-win situation". They fully have the option to go against their own values if they feel it appropriate. I wouldn't call infiltrating the enemy, learning their plan, and saving hundreds of lives at the expense of having to set your personal values aside to be a "no-win". If there are mechanical penalties for going against their view, such as a 3.5 Druid losing their magic for 24 hours if they don metal, they can opt to deal with that for the greater good. That's not true with a DM ruling a player won't do something, and is the only case in which a "no-win situation" can possibly happen (not including something silly like a DM offering two death doors), which is why it's a bad concept to bring to the table.
It's funny how half the people defending the "no armor" bit are saying "It's the rules, druids aren't allowed metal armor, stop trying to break the rules," and the other half are saying, "It's just a story restriction, it's not part of the rules, so why are you complaining?"
And that's exactly a major problem with the wording and placement of the rule. People asked the game developers because of the confusion caused by the clause being slid into their proficiencies without explanation. The wording and placement makes people assume it's a mechanical restriction, but the Sage Advice says it's just lore, Druids don't lack the ability to wear it, and that nothing prevents a Druid from doing so besides their own choice not to.
The Sage Advice definitely makes it clear to me that the restriction is not one that has to be followed, because characters are allowed to make choices that go contrary to their tenets, but because it's listed in proficiencies people still stand by the fact that it is a mechanical restriction and that Druids will stop being Druids if they don't agree. Apparently they need to do more to make the point clear, as people don't even agree on what the Sage Advice is trying to say, but it seems they simply don't want to expand upon it.