What should be done regarding a person who lied and then apologized, then accused me of being insane after I accepted the apology, and when I confronted him about it, he denied it and swore by Allah that he "does not remember" his action, and claimed that I am a liar, and refused to swear by the words "I did not do it"?

Swearing by Allah falsely is an oath of perjury (al-yamin al-Ghamus), which is a major sin. What is obligatory for whoever commits it is to repent to Allah and resolve not to repeat it. There is no expiation for an oath of perjury according to many scholars, because it is too great for an expiation to atone for it. If a person swears that he does not remember something, he must be believed as long as he has sworn by Allah, and he is truthful in his oath even if he has forgotten what he did. However, if he remembers and swore falsely, then it is an oath of perjury. A Muslim should be characterized by forbearance, forgiveness, and overlooking wrongdoers, and should know that Satan seeks to sow discord among Muslims.