All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
It is impermissible for the Muslim to ask someone to kill him under any circumstances. If he does so and that person kills him, then he has caused himself to be killed, but he did not kill himself and it is not considered suicide. According to Mu‘jam Lughat Al-Fuqahaa, the Arabic term for suicide: "Intihaar" is derived from the verb "Nahara", which means: slaughtered/cut the throat. In other words, it means killing oneself with some tool. Mu‘jam Al-Lughah Al-‘Arabiyyah Al-Mu'aasirah defines suicide as follows: killing oneself consciously or unconsciously, or the deliberate act of taking one's own life.
If the person who is asked to kill him is crazy, for example, and he is not accountable for his actions under the Islamic Shariah and would most likely carry out what he was asked to do, then in this case ordering him to kill becomes close to suicide.
Allah knows best.