You only can have one unique ID per page. Change those IDs to classes, and change the jQuery selectors as well.
Also, move the .on() outside of the .click() function, as you only need to set it once.
http://jsfiddle.net/samliew/3AJcj/2/
$(document).ready(function(){
$(".addCF").click(function(){
$("#customFields").append('<tr valign="top"><th scope="row"><label for="customFieldName">Custom Field</label></th><td><input type="text" class="code" id="customFieldName" name="customFieldName[]" value="" placeholder="Input Name" /> <input type="text" class="code" id="customFieldValue" name="customFieldValue[]" value="" placeholder="Input Value" /> <a href="javascript:void(0);" class="remCF">Remove</a></td></tr>');
});
$("#customFields").on('click','.remCF',function(){
$(this).parent().parent().remove();
});
});
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