The server has held up pretty well, since the installation of varnish. Based on this wiki post, I added the following to /etc/varnish/default.vcl

<pre>
# Drop any cookies sent to WordPress.
sub vcl_recv {
        if (!(req.url ~ "wp-(login|admin)")) {
                unset req.http.cookie;
        }
}
# Drop any cookies WordPress tries to send back to the client.
sub vcl_fetch {
        if (!(req.url ~ "wp-(login|admin)")) {
                unset beresp.http.set-cookie;
        }
}

I think the comments are pretty self explanatory.