The issue is caused by the fact that you are applying the tooltip either way, irrespective of if the button is visible at all. To go around that, you'll have to first give an id
to your buttons
...
<button id="small" type="button" class="btn btn-default tooltip-btn visible-xs visible-sm" data-toggle="tooltip" data-placement="bottom" title="Small Screen Tooltip">visible-xs, visible-sm</button>
<button id="medium" type="button" class="btn btn-default tooltip-btn visible-md visible-lg" data-toggle="tooltip" data-placement="bottom" title="Large Screen Tooltip">visible-md, visible-lg</button>
... then check for their visibility in js as follows:
if ($('#small').is(':visible'))
$('#small').tooltip(opts).tooltip("show");
if ($('#medium').is(':visible'))
$('#medium').tooltip(opts).tooltip("show");
I updated your PasteBin here: http://jsbin.com/UFoRIYex/509/edit