<pre>
<?
/*$date = new DateTime();
print_r($date->format('Y-m-d\TH:i:sP'));*/

$name_portal = 'uhykhjk';
$key = 'khj7687u';

$params = [
    'filter' => [
        'AUTHOR_ID' => '72',
        'RESPONSIBLE_ID' => '560'
    ],
    'start' => 0
];

function getAllActivity($name_portal, $key, array $params) {
    $queryUrl = 'https://'.$name_portal.'.bitrix24.ru/rest/1/'.$key.'/crm.activity.list.json';
    $queryData = http_build_query($params);

    $curl = curl_init();
    curl_setopt_array($curl, array(
     CURLOPT_SSL_VERIFYPEER => 0,
     CURLOPT_POST => 1,
     CURLOPT_HEADER => 0,
     CURLOPT_RETURNTRANSFER => 1,
     CURLOPT_URL => $queryUrl,
     CURLOPT_POSTFIELDS => $queryData,
    ));

    $result = curl_exec($curl);
    curl_close($curl);

    $result = json_decode($result, 1);
    return $result;
}

$firstArr = getAllActivity($name_portal, $key, $params);

$total = $firstArr['total'];
$start = 50;
while($start < $total){
   $firstArr['start'] = $start;
   $firstArr['result'][] = getAllActivity($name_portal, $key, $params)['result'];
   //тут обработка $arRes и в конце
   $start = $start + 50;
}

print_r($firstArr);

?>