HTML ve JS nizi Bu Şekilde Düzenleyiniz Hocam.. js link kısımlarını kendi linklerinize göre düzenleyin..
Kolaygelsin..

<?php	

		$tweet = $connection->get("https://api.twitter.com/1.1/statuses/user_timeline.json?screen_name=".$username."&count=10");
		foreach ($tweet as $ayikla)
		{
		echo'<article class="box is-post">';
		echo '<b>'.$ayikla->text.'</b>'; //Tweet
		$tweetidsi = $ayikla->id_str;
		//echo $tweetidsi;
		echo'
		<form  method="post" action="index.php?islem=fav" ><input name="favid" type="text" class="m-wrap" value="'.$tweetidsi.'" /> <input name="fav" type="button" class="button"  value="10 Favori Ekle"/></form>
        <form method="post" action="index.php?islem=rt" ><input name="rtid" type="text" class="m-wrap"  value="'.$tweetidsi.'" /> <input name="rtw" type="button" class="button"  value="10 Retweet Ekle"/></form>
    ';
		
		}

?>

	  $('.button').click(function(){
		islem=$(this).attr('name');
		twit_id=$(this).parent().find('.m-wrap').val();
		
		if(islem=="fav")
		 {
		  adres="favori.php";
		 }else{
		 adres="retwit.php";
		 }
	
		 yolla="twit_id="+twit_id;
		 
		 $.ajax({
		   type: "post",
		    url: adres,
		    data :yolla,
		    success : function(cevap){
		   alert(cevap);
		      //$("#goster").html(cevap);
		  }
		 });

	  });