In an Islamic state governed by sharia law, if a person finds Islam no longer believable and defects from the Islam religion, not performing any of the spiritual worship requirements such as Salavat 5 times a day or fasting during Ramadan, not believing in angels, jinns or demons, is that person supposed to be killed or prosecuted in any way?
The verse 5:32 makes it clear that no one can be killed except in two cases:
...whoever killed a human being, except as a punishment for murder or for spreading corruption in the land, shall be regarded as having killed all mankind... [5:32]
Similarly, a state can enforce only two religious directives:
But if they repent, and take to prayer regularly and pay the alms, then let them go their way. [9:5]
The combined result of the above two verses is that the state can punish but not kill a person for leaving salah and zakah. This, of course, is for a person who still claims to be a Muslim.
If a person renounces Islam but is not guilty of spreading corruption in the land then he cannot be killed. As you see in the following verse a similar case is being discussed but God does not prescribe any worldly punishment for him:
As for those who come to believe, and then deny the truth, and again come to believe, and again deny the truth, and thereafter grow stubborn in their denial of the truth -- God will never forgive them, nor will He guide them. [4:137]
Generally, killing the Kafir/Moshrik in the battle field (i.e. the Kafir who is fighting and violating Islam and Muslims), but at the common situation it is related to the Hokm (ruling) of Islamic court. I.e. this Hokm, ought to be performed if the Islamic court recognized that murdering a person (Kafir or even Muslim) is Wajib (mandatory).