Hi nauna,

Make the scroll code as function.

Call the function after UpdatePanle partial postback.

<asp:UpdatePanel runat="server" UpdateMode="Conditional">
    <ContentTemplate>
        <div class="chat_container">
            <asp:Timer ID="timer" runat="server" Interval="2000" OnTick="Timer1_Tick">
            </asp:Timer>
            <asp:ListView ID="ListView1" runat="server">
                <ItemTemplate>
                    <div class="all-msgs d-inbox">
                        <asp:LinkButton ID="LinkButton1" runat="server">View</asp:LinkButton>
                        <asp:Label ID="Label3" runat="server" Text='<%#Eval("Id") %>' CssClass="badge pull-right theme-bg"></asp:Label>
                    </div>
                </ItemTemplate>
            </asp:ListView>
            <asp:Label ID="lblID" runat="server" />
            <asp:HiddenField ID="hfQuerStringID" runat="server" />
            <asp:HiddenField ID="hfSelectedLabel" runat="server" />
            <input type="hidden" id="div_position" name="div_position" />
        </div>
    </ContentTemplate>
    <Triggers>
        <asp:AsyncPostBackTrigger ControlID="timer" EventName="Tick" />
    </Triggers>
</asp:UpdatePanel>
<style type="text/css">
    .chat_container {
        overflow-y: auto;
        display: block;
        max-height: 650px;
        height: 100%;
        min-height: 600px;
        font-family: 'Roboto', sans-serif;
    }

    .selected {
        background-color: #f44336;
        color: white;
        text-align: center;
        text-decoration: none;
    }
</style>
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript">
    $('body').on('click', '[id*=LinkButton1]', function () {
        $('[id*=Label3]').removeClass();
        $('[id*=Label3]').addClass('badge pull-right theme-bg');
        var id = $(this).closest('div').find('[id*=Label3]').html();
        var oldURL = window.location.protocol + "//" + window.location.host + window.location.pathname;
        if (history.pushState) {
            var newUrl = oldURL + "?ID=" + id;
            window.history.pushState({ path: newUrl }, '', newUrl);
            $('#hfQuerStringID').val(id);
        }
        scroll();
        $('#hfSelectedLabel').val($(this).closest('div').find('[id*=Label3]').attr('id'));
        $(this).closest('div').find('[id*=Label3]').addClass('selected');
        return false;
    });
    $(function () {
        scroll();
    });

    function scroll() {
        var div = $('.chat_container');
        var div_position = document.getElementById("div_position");
        var position = parseInt('<%=Request.Form["div_position"] %>');
        if (isNaN(position)) {
            position = 0;
        }
        div.scrollTop = position;
        div.onscroll = function () {
            div_position.value = div.scrollTop;
        };
    }
    //On UpdatePanel Refresh.
    var prm = Sys.WebForms.PageRequestManager.getInstance();
    if (prm != null) {
        prm.add_endRequest(function (sender, e) {
            if (sender._postBackSettings.panelsToUpdate != null) {
                scroll();
                var id = $('#hfSelectedLabel').val();
                $('#' + id).removeClass();
                $('#' + id).addClass('selected');
            }
        });
    };
</script>