You can approximate this functionality with a BugMonkey script, like
name: Update List
description: Add update functionality
author: Daniel LeCheminant, Michel de Ruiter
version: 1.0.1.0
js:
$(function() {
var storage = window.localStorage;
if (!storage || $("#bugGrid").size() == 0)
return;
var getVal = function(key, def) {
return (storage && storage[key]) || getCookie(key) || def;
};
var setVal = function(key, val) {
storage ? storage[key] = val : setCookie(key, val);
};
var update = function() {
if ($(".rCB:checked").length || $("#searchFor").val().length) {
reset();
} else {
Info.show("Updating...")
location = "?pg=pgList";
}
};
var fixMins = function(sMins) {
return Math.max(parseInt(sMins) || 5, 1);
};
var getMins = function() {
return fixMins(getVal("cMinsRefresh", ""));
};
var timeout;
var stop = function() {
timeout = clearTimeout(timeout);
};
var reset = function(fStart) {
if (timeout || fStart) {
stop();
timeout = setTimeout(update, getMins() * 60 * 1000);
}
};
$(document).bind("click mousedown keydown", function() { reset(); });
$(window).bind("scroll", function() { reset(); });
var fAutoRefresh = getVal("fAutoRefresh", false) == "true";
var span = $("<span class='update'>").insertAfter("#versionFooter .version:last");
$('<input type="checkbox">')
.attr("checked", fAutoRefresh ? "checked" : null)
.click(function() {
(fAutoRefresh = $(this).is(":checked")) ? reset(true) : stop();
setVal("fAutoRefresh", fAutoRefresh);
}).appendTo(span);
span.append("Auto update every <input/> minutes")
.find("input:last").val(getMins())
.keyup(function() {
setVal("cMinsRefresh", fixMins($(this).val()));
reset();
})
.blur(function() {
$(this).val(getMins());
});
reset(fAutoRefresh);
});
css:
span.update { float: right; }
span.update > input[type='checkbox'] + input { width: 20px; text-align: right; }
... which will give you a little checkbox and input at the bottom of your filter:

Some important notes/caveats:
- It actually does do a page reload :-(
- The update will behave exactly as if you clicked the "List Cases" link on FogBugz, so if you've changed your current filter in another window, you'll see that filter after the refresh.
- The script tries to stay out of your way, so if won't ever update if you've checked some cases or entered some search terms in the search box.
- If you click, type, or scroll the window, the timeout will be reset. (Again, the reduce the chances that the page will update when you're in the middle of doing something)
- The refresh settings are per-browser, using local storage or cookies.