The 'Odoo Support' is defined in an XML file and JavaScript file so you can easily comment them out.
1. Go into the folder im_odoo_support under addons and then go to static/src/xml and open the file im_odoo_support.xml You'll see the following code and then you should just comment out the li class for Odoo support as I did:

<?xml version="1.0" encoding="UTF-8"?>

<templates xml:space="preserve">
    <t t-extend="UserMenu">
        <t t-jquery=".dropdown-menu li:eq(3)" t-operation="after">
    <!--
            <li class="odoo_support_contact">
                <a data-menu="odoo_support" href="#">Odoo Support</a>
            </li>-->
        </t>
    </t>
</templates>

2. Go up some levels again, untill you are back in the folder im_odoo_support and then open the folder views. You'll find another xml file named im_odoo_support.xml. You'll see the following code. Just comment out the full template as I did:

<?xml version="1.0" encoding="utf-8"?>
<openerp>
    <data>

       <!-- <template id="assets_backend" name="im_odoo_support assets" inherit_id="web.assets_backend">
            <xpath expr="." position="inside">
               <script type="text/javascript" src="/im_odoo_support/static/src/js/im_odoo_support.js"></script>
                <script type="text/javascript">
                    (function() {
                        openerp.im_odoo_support.support = new openerp.im_odoo_support.OdooSupport(
                            "<t t-esc="request.session.login"/>",
                            "<t t-esc="request.registry['ir.config_parameter'].get_param(request.cr, request.uid, 'database.uuid')"/>"
                        );
                    })();
                </script>
            </xpath>
        </template>-->

    </data>
</openerp>

3. Reload and update your module and you no longer see the Odoo support in your chat.
Example: