Try this:
You have to wrap the code inside the UpdatePanel
<form id="form1" runat="server">
<div style="min-height: 700px;">
<asp:ToolkitScriptManager ID="ToolkitScriptManager1" runat="server">
</asp:ToolkitScriptManager>
<asp:UpdatePanel ID="UpdatePanel1" runat="server" UpdateMode="Conditional">
<ContentTemplate>
<asp:Button ID="Button1" runat="server" Text="Click here to show the modal" />
<asp:TextBox ID="TextBox1" runat="server" AutoPostBack="true" ontextchanged="TextBox1_TextChanged"
></asp:TextBox>
<asp:ModalPopupExtender ID="ModalPopupExtender1" BackgroundCssClass="ModalPopupBG"
runat="server" CancelControlID="btnCancel" OkControlID="btnOkay" TargetControlID="Button1"
PopupControlID="Panel1" PopupDragHandleControlID="PopupHeader" >
</asp:ModalPopupExtender>
<div id="Panel1" style="display: none;" class="popupConfirmation">
<div class="popup_Container">
<div class="popup_Titlebar" id="PopupHeader">
<div class="TitlebarLeft">Popup Header</div>
<div class="TitlebarRight"></div>
</div>
<div class="popup_Body">
<asp:Repeater ID="Rptr_popup" runat="server">
<ItemTemplate>
<%-- <asp:LinkButton ID="lnkbtn_Contperson" runat="server" Text='<%# Eval("contname")%>'></asp:LinkButton>--%>
<asp:RadioButton ID="RadioButton1" runat="server" Text='<%# Eval("contname")%>' TextAlign="Left"/>
</ItemTemplate>
</asp:Repeater>
</div>
<div class="popup_Buttons">
<input id="btnOkay" value="Done" type="button" />
<input id="btnCancel" value="Cancel" type="button" />
</div>
</div>
</div>
</ContentTemplate>
<Triggers>
<asp:AsyncPostBackTrigger ControlID="TextBox1" />
</Triggers>
</asp:UpdatePanel>
</div>
</form>