Our zabbix is using http authentication with apache and mod_auth_mysql using zabbix's internal users table (to provide easier scripted access to zabbix). Currently zabbix disables password changing if authentication type isn't ZBX_AUTH_INTERNAL. This can be solved with following modification in include/forms.inc.php:
replace if(ZBX_AUTH_INTERNAL == $auth_type) with if(ZBX_AUTH_INTERNAL == $auth_type || ZBX_AUTH_HTTP == $auth_type) on line 765 (1.6.1).

It would be nice if there was some sort of checkbox in administration->authentication->http that would allow changing the password in such cases.