strtotime returns int and the second parameter (if used) should be a valid timestamp. Also if you want to get the date string you can use date function
$start_date = strtotime('2013-12-04 12:00');
for ( $i = 1; $i<9; $i++){
$start_date= strtotime('+1 week', $start_date);
$arr = array('id' => $row["course_id"],
'title' => $row["course_name"],
'start' => date('Y-m-d h:i:s',$start_date),
'allDay' => false
);
array_push($result, $arr);
}