The answer to this question depends greatly upon the standpoint from which it is addressed.
Legally, for example, we have no obligation to examine our beliefs in the first place. If we do examine our beliefs, we have no obligation to do so based on ethics or morals. Our only obligation is to refrain from breaking laws.
Similarly, there are some philosophical points of view that do not emphasize any need to examine our beliefs from a moral or ethical standpoint. One example of such a point of view is utilitarianism. Utilitarianism does not ask us to examine what we believe. We are free to believe what we want so long as we act in ways that increase the amount of happiness that is experienced by the people around us.
By contrast, there are schools of thought that are more likely to say that we must examine our beliefs from a moral or ethical perspective. Kant’s philosophy is one such perspective. To Kant, we need to have rationales for our behavior. We need to know what sorts of maxims guide our actions so that we can know whether we would will them to become universal laws. From this sort of perspective, we do have an obligation to examine our beliefs morally.