I've always loved the instruction about this in Proverbs 26: 4-5.

I've always loved the wisdom behind this (sadly, not that I keep it all the time). Even those of us that take the Hebrew Bible as from God need to remember that the proverbs are wisdom literature. So there's a subjective element.

What I get out of this brilliant back-to-back juxtaposition is that we should argue our case for the sake of the fool as long as it doesn't suck us into that manner of thinking.

Basically, I agree with @Nadia. The difficulty is in seeing when someone is innocently ignorant or willfully ignorant. I may not be great at interpreting my own thoughts, but its a total guessing game discerning someone else's. Using the advice in these proverbs, we don't even need to know that answer though. We can try to help until the moment that we see ourselves getting sucked into emotional or irrational arguments.