It was bug in the Firefox, and now works as expected. Here is markup for testing:
<script runat="server">  
    protected void RadTextBox1_TextChanged(object sender, EventArgs e)
    {
        newText.Text = RadTextBox1.Text;
    }  
</script>
<asp:Label runat="server" ID="newText"></asp:Label>
<telerik:RadTextBox runat="server" ID="RadTextBox1" AutoPostBack="true" OnTextChanged="RadTextBox1_TextChanged"></telerik:RadTextBox>
<asp:Button Text="submit" runat="server" />