Why does Montresor feel he has the right to take justice into his own hands?
Montresor does not necessarily feel he has a right to take justice into his own hands. He knows he is deliberately and intentionally committing a serious crime, and he doesn't want to get caught. He does not have to be insane to do what he does to Fortunato. People commit murders all the time, as police statistics show. Most of them probably do not assume they have a right to kill someone. They may be motivated by rage or by greed or some other strong emotion. If they were actually insane, they probably wouldn't be punished. If they had a "right," then they probably wouldn't be punished either. Montresor would be punished under Italian law if the authorities found out that he had murdered Fortunato. He was well aware of that, and he took extreme precautions to get away with what he knew was a capital crime.
Montresor is insane, but you should also consider his family's motto that no one harms a Montresor without being punished. Even his family's crest is of a snake biting a heel, so acts of revenge run in his family. We don't know whether other people in his family took revenge to the extent that Montresor does against Fortunato, but he believes he has the right to take justice in his own hands partly because of his family's name, but mostly because of his madness. He has taken revenge to the extreme, and the reader isn't sure whether Fortunato even did anything to Montresor. Because he's insane, Montresor may have just imagined that Fortunato had insulted him.