Not all ideas and beliefs are equally reasonable and valid. I think it is good to scrutinize our own beliefs and ideas, test them against reason and experience, and change them if need be. And I think others should also be encouraged to do that.
However, we are not all the same. People have different brain structures and some people are stronger in analytical things, while others are stronger in intuitive things, or whatever. Some people just do not have it in their nature to question every thought and idea like we do. To someone like me, who is very analytical and scrutinizes my own thoughts, that can be very frustrating! But I try to step back and realize that not everyone's brain is wired up the way mine is. They may be strong where I am weak, and vice versa.
I think that what we must respect is not the ideas and beliefs themselves, but the right of each person to hold those beliefs. From where they stand, the belief looks right, or else they wouldn't hold that belief. We have to respect that each person may be at a different level, and respect where that person currently is in life. I don't think we need to respect refusal to question. But if after encouraging someone to think a little deeper about their beliefs, they refuse to do so, I think we have to finally just step back and let them be.
And I try to respect them even for that, by having compassion on them - because there must be some emotional reason or something that makes them fear losing a cherished belief. And we should have compassion on them, since they have a deep emotional fear of some sort, or whatever the case may be. In other words, they have a reason for holding on to an unreasonable belief! And I think we should respect that situation they are in.