I can not think of any beliefs that are not worth examining other than those which fall in the spiritual and religious realm such as believing in the possibility of miracles. However, I do think many of the beliefs held in religious institutions should be critically studied because those beliefs affect many people's daily lives (i.e., beliefs on gay marriage, gender equality, etc.). 

To give a more general example, consider Immanuel Kant's "categorical imperative" which states that, "Act only according to that maxim by which you can at the same time will that it should become a universal law." This sounds well and good, but it does meet with problems. Let's include "you should never lie" as a categorical imperative. What if there is someone at the door whom you know is there to kill one of your friends in the house. If "you should never lie," you can't lie and say that friend isn't there. This shows that there are problems with fundamental beliefs and therefore even something as logical as "you should never lie" must be put under scrutiny in hypothetical and real world situations. 

What Kant was suggesting was that if everyone did not lie, did not steal, did not kill, etc., then we'd be living in a rational utopia where the murderer-at-the-door situation would never occur. However, since we don't live in a perfect world, some beliefs just can't be always upheld. Kant was trying to establish a moral framework in order to get closer to that world where people are rational and responsible. 

If you believe without examining, you will always feel confident and comfortable.  Ignorance is bliss, after all.  However, you also have a strong chance of being wrong.  Either someone has told you something and you believe it without question or you made something up.  Either way, chances are you are making an erroneous assumption and might make bad choices based on your bad assumption.

Aside from religious beliefs, which are clearly a matter of faith, not logic or evidence, if we are not capable of examining our beliefs, subjecting them to scrutiny, I do not think our belief are worth very much.  One way to do this is to subject one's belief to an extreme hypothetical or real application.  For example, I have a strongly held belief in the freedom of speech, as guaranteed by the First Amendment of the United States Constitution.  As extreme cases of this freedom have cropped up over the years, for example, Neo-Nazis marching in Skokie, Illinois, pro-life protesters at abortion clinics, and that church group that protests against homosexuality at private funerals, I have asked myself how strong my belief really is, whether or not it can hold against "speech" that is so offensive to me.  So far, my belief has remained strong, in spite of the distasteful nature of some speech.  It is always possible to test one's beliefs this way, either in a real world situation or in a thought experiment. 

 What is the importance of subjecting our beliefs to critical scrutiny?

To begin, we might say that testing your beliefs critically is likely to do one of two things: strengthen your current beliefs or identify beliefs that are characterized by fallacy, flaws in logic, or consistency, etc. 

This can be seen as a process of vetting. The worthy and strong beliefs are maintained and even become stronger (because they have been assessed, tested and proven in some way) and the weaker beliefs are adjusted or eliminated. 

To use a metaphor here - If your intellectual life is an automobile and your beliefs are the tires, you would certainly want to inspect your tires once in a while. 

You check the tires, not because you don't believe in them in the first place, but because it is practical to make sure that over time and wear they haven't begun to show weaknesses that could lead to a blowout.