whileの中で｢$tameru_tutor = array();｣とやっているので､毎回初期化している感じですね｡
whileの外に出してしまえばいいのでは？

ご回答､ありがとうございます｡具体的にはどんなﾌﾟﾛｸﾞﾗﾑになりますでしょうか?

$cursor = -1;
$tameru_tutor = array();
while($cursor != 0){

 $vRequest_tutor = $twObj_tutor->OAuthRequest("https://api.twitter.com/1.1/followers/list.json","GET",array('screen_name' => 'XXXX','cursor' => $cursor,'skip_status' => 'true','include_user_entities' => 'false' ));

 $temp = json_decode($vRequest_tutor,true);

 if (isset($temp['next_cursor_str'])) {
 $cursor = $temp['next_cursor_str'];

 $tameru_tutor[] = $temp;
 unset($temp);
 } else {
 $cursor = "0";
 }
}//end of while

$tameru_tutorは二次元のような感じになりますので
foreach ($tameru_tutor as $temp) {
foreach ($temp["users"]) as $user) {
echo $user["name"];
}
}
或いは$temp["users"]だけでよいのであれば
$tameru_tutor[] = $temp;
を
$tameru_tutor = array_merge($tameru_tutor, $temp["users"]);
とすれば
foreach ($tameru_tutor as $user) {
echo $user["name"];
}
でｱｸｾｽできると思います｡