Loyalty by its definition is complete faith in someone or in some cause. By this definition alone we cannot understand the detrimental side of loyalty. In its simplest form, it is observed as a virtue that reassures us that amidst the vices of man, there is some good. In most things that bring immediate good, we tend to not look for the harm that it could cause in extreme cases. Thus we always tend to ignore the detrimental aspect to it. It is possible to discern the detrimental side to it if we look at loyalty in its extremes.
The serious leeward side of loyalty is the blindness it brings. A complete lack of cognitive behaviour because of peoples near fanatical loyalty to a faith, are the causes of destruction and devastation today. A more relevant example is not available to confirm the veracity in the fallibility of blind faith. It is a characteristic of terrorism. The fanaticism and insane loyalty to a cause, where they pledge allegiance to their faith, and see it fit for numerous atrocities to be committed. All attacks on humanity or prevailing harmony, which are globally condemned, are all acts conducted by such intense loyalty to a cause. This loyalty leads one to ignore all reason and then perform dastardly acts.
Intense loyalty is not only harmful to others, it can very often be, in fact more often be harm to ones-self. When it comes to loyalty to God for example, it brings with it blindness; a kind of fanaticism again but which harms the self. When among the religious, become ill, if they believe that god or his miracles will alleviate them from their disease, and wait for another miracle they are doomed. People are believed to have refused medical attention to treat their cancer, because they believe that God is the one who put in a cancer, he is the one who will remove it.
These are the more severe effects of loyalty which make it as detrimental at least as much as it is beneficial As it is so with everything, it is so with loyalty as well, in its excess it is extremely harmful. Everything must be reasoned out. Even loyalty as godly as it maybe, sometimes has its shortcomings.