How to make TinyMCE work inside an UpdatePanel?

Ok, your problem is two fold. Stefy supplied you with part of the answer, which is you have to initialize TinyMCE on the postback by registering startup script like so:

using System.Web.UI;

namespace TinyMCEProblemDemo
{
    public partial class EditorClean : UserControl
    {
        protected void Page_Load(object sender, System.EventArgs e)
        {                
              ScriptManager.RegisterStartupScript(this.Page, 
                  this.Page.GetType(), mce.ClientID, "callInt" + mce.ClientID + "();", true);
        }
    }
}

The second problem you have is with your implementation of a custom control. Designing custom controls is out of scope of this answer. Google can help you there.

You have multiple instances of your control on the page which can cause you issues with script, as it get rendered multiple times. This is how I modified your markup to solve your issue(notice dynamic naming of your script functions, custom controls should be self contained and mode: "exact" on the tinyMCE.init):

<%@ Control Language="C#" AutoEventWireup="true" CodeBehind="EditorClean.ascx.cs"
    Inherits="TinyMCEProblemDemo.EditorClean" %>
<script type="text/javascript" src="Editor/tiny_mce.js"></script>

<script type="text/javascript">
    function myCustomCleanup<%= mce.ClientID%>(type, value) {
        if (type == "insert_to_editor") {
            value = value.replace(/&lt;/gi, "<");
            value = value.replace(/&gt;/gi, ">");
        }
        return value;
    }
    function myCustomSaveContent<%= mce.ClientID%>(element_id, html, body) {
        html = html.replace(/</gi, "&lt;");
        html = html.replace(/>/gi, "&gt;");
        return html;
    }

    function callInt<%= mce.ClientID%>() {

        tinyMCE.init({
            mode: "exact",
            elements: "<%= mce.ClientID%>",
            theme: "advanced"