How can I remove the welcome message generated by the New User Approve plugin?

The second argument should be a function callback, not the actual message. You can remove the message completely by pasting this code into your theme’s functions.php file:

add_filter( 'new_user_approve_welcome_message', 'wpse_76823_login_message' );

function wpse_76823_welcome_message( $message ) {
    return '';
}

Or, you can replace the message by using this code:

add_filter( 'new_user_approve_welcome_message', 'wpse_76823_login_message' );

function wpse_76823_welcome_message( $message ) {
    return "This is my new message";
}

If you need more help, post a comment below.