Is the meaning of 把人的牙笑掉 exactly the same as 'laugh your head off'?
Is "laughing one's head off" the exact equivalent of 把人的牙笑掉? Of course not.



The original line from《紅樓夢》goes like this:


  探春黛玉都笑道：「誰不是玩？難道我們是認真做詩呢？要說我們真成了詩，出了這園子，把人的牙還笑掉了呢！」


David Hawkes, published, translation in The Story of The Stone Vol. II reads like this:


  ‘Good heavens! we only write for the fun of it ourselves,’ said Tan-chun. ‘You surely don’t imagine that what we write is good poetry? If we set ourselves up to be real poets, people outside this Garden who got to hear of it would laugh so loud that their teeth would drop out!’


H. Bencraft Joly's, published, translation in The Dream of the Red Chamber reads:


  Tanchun and Daiyu both smiled. “Who doesn’t go in for these things for fun?” they asked. “Is it likely that we improvise verses in real earnest? Why, if anyone treated our verses as genuine verses, and took them outside this garden, people would have such a hearty laugh at our expense that their very teeth would drop.”



Both of these, published, translations opt for "teeth dropping out" from laughter.


Yang Hsien-Yi and Gladys Yang's, published, translation in A Dream of Red Mansions writes:


  “Who's not doing it for fun?” countered Tan-chun and Tai-yu. “We don't write seriously either. If we really set up as poets, people outside the Garden would split their sides laughing.”



Here it is translated “split their sides laughing.” 




Here are some other sample translations from 91dub:


  布兰丁斯城堡 第一季
  people will laugh.
  人们准会笑掉大牙





  天堂真的存在
  will get me laughed out of this town.
  只会让人们笑掉大牙





  摩登家庭 第七季
  sitting at their lawn chairs, laughing their asses off
  坐在躺椅上笑掉大牙





  绝望主妇 第八季
  Besides trying not to laugh my ass off?
  除了忍住不让自己笑掉大牙外吗





  暂告安全
  When we heard what Docheski did to you,I mean,I had to laugh.
  听说多切斯基干的事 我都要笑掉牙了




It seems it can also be interchanged with 下巴:


  达·芬奇的恶魔 第一季
  He'll laugh his arse off with the rest of them.
  他会跟其他人一样笑掉下巴




"Laugh your head off" is, though, a great translation, but it could be interpreted in a number of ways.