I had the same problem using Bootstrap V3. I followed the tutorial on the official documentation, but couldn't get the sidebar to collapse by default on mobiles.
This comes from a bug in Bootstrap V3 source files (custom version generated on 2013/09/05). Just load the following CSS right after the original bootstrap.css :
.collapse{display:none}.collapse.in{display:block}.collapsing{position:relative;height:0;overflow:hidden;-webkit-transition:height .35s ease;transition:height .35s ease}@media screen and (min-width:768px){.nav-collapse.collapse{display:block!important;height:auto!important;overflow:visible!important}}(Those styles come from a previous version of the original Bootstrap V3 source).
Otherwise, my only mistake was to initialize the collapse plug-in in my HTML page. So if you've got something like this in your HTML source code :
<script type="text/javascript"> $(document).ready(function(){ $('.collapse').collapse(); });</script>...then just remove it and the collapse plug-in will work like a charm.