We need first to understand the reasons behind a person who claims that he wants to leave Islam. Usually such persons suffer from misunderstanding Islam, or having bad reaction to bad actions of some so-called Muslims.
We need to be sure that he had full access to real knowledge about Islam.
We are responsible to provide him with the facts from Quran and authentic Hadeeths from the Prophet (SAWA) and his Holy Progeny Ahlul Bayt (AS).
There is no question of punishment before being sure that all. facts are clear to him and he is insisting to damage the faith of others by claiming leaving the Religion of Islam.
In this time when the Islamic justice is not implemented and there is no government of the Infallible Imam, we have no right to punish such person. It is for the Infallible Imam or his deputy to decide not the public.