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.