So I think I figured this out: my original markup relied solely on the data-target element, but that is apparently not enough. Safari (on iPhone) seems to also need the href attribute (which really should be there on an <a> anyway. So this works:
<a data-toggle="collapse" data-target="#collapse1" href="#collapse1">
<i class="pull-right icon-chevron-right mobile-nav-icon"></i>
</a>
But this does not:
<a data-toggle="collapse" data-target="#collapse1">
<i class="pull-right icon-chevron-right mobile-nav-icon"></i>
</a>