There is no way to logout with JSON API.

##Attempt 1
##Request

{"jsonrpc":"2.0","method":"user.logout","params":"dad016d2d1fc782ccdfec648327f96a1","auth":"dad016d2d1fc782ccdfec648327f96a1","id":9}

##Response
{"jsonrpc":"2.0","error":

{"code":-32602,"message":"Invalid params.","data":"JSON-rpc params is not an Array."}

,"id":9}

##Attempt 2
##Request

{"jsonrpc":"2.0","method":"user.logout","params":["f463f3fc908972b66c0662b47363b1da"],"auth":"f463f3fc908972b66c0662b47363b1da","id":8}

##Response
{"jsonrpc":"2.0","error":{"code":-32602,"message":"Invalid params.","data":"Cannot logout.","debug":[

{"file":"\/usr\/share\/zabbix\/api\/classes\/CUser.php","line":884,"function":"exception","class":"CZBXAPI","type":"::","args":[100,"Cannot logout."]}

,

{"function":"logout","class":"CUser","type":"->","args":[["f463f3fc908972b66c0662b47363b1da"]]}

,{"file":"\/usr\/share\/zabbix\/api\/rpc\/class.czbxrpc.php","line":106,"function":"call_user_func","args":[[{},"logout"],["f463f3fc908972b66c0662b47363b1da"]]},

{"file":"\/usr\/share\/zabbix\/api\/rpc\/class.czbxrpc.php","line":64,"function":"callAPI","class":"czbxrpc","type":"::","args":["user.logout",["f463f3fc908972b66c0662b47363b1da"]]}

,

{"file":"\/usr\/share\/zabbix\/api\/rpc\/class.cjsonrpc.php","line":66,"function":"call","class":"czbxrpc","type":"::","args":["user.logout",["f463f3fc908972b66c0662b47363b1da"],"f463f3fc908972b66c0662b47363b1da"]}

,

{"file":"\/usr\/share\/zabbix\/api_jsonrpc.php","line":54,"function":"execute","class":"CJSONrpc","type":"->","args":[]}

]},"id":8}