Always when i cant find the right elements in a big array, i use print_r() to print it and find it.
This will be your solution:
<?php
$JSON = file_get_contents("http://gdata.youtube.com/feeds/api/videos?&v=2&q=dance&alt=json");
$JSON_Data = json_decode($JSON);
foreach($JSON_Data->feed->entry as $video):
echo $video->id->{'$t'}."<br/>";
endforeach;
?>