There's no convenient way of doing this in a totally cross browser manner.  Firefox, Chrome and Safari all support the DOMSubtreeModified event which will fire for any modifications to the DOM inside a particular element.  The aforementioned browsers and Opera support DOMNodeInserted and DOMNodeRemoved, which will also do the job. 
$("#placesNewCheckinContainer").bind("DOMNodeInserted DOMNodeRemoved", function () {
    alert("Something happened inside #placesNewCheckinContainer");
});
Unfortunately, IE doesn't support any of those events.  It does have its own event that can partially do the job, however - onpropertychange.  This can notify you when the value of innerHTML changes, but only for modifications to direct children, not all descendants.