You can use <asp:Literal to output a clean string on your page.
Do this:
<iframe width="560" height="315" src="http://www.youtube.com/embed/<asp:Literal ID="keyOfVideo" runat="server"></asp:Literal>" frameborder="0" allowfullscreen></iframe>
CodeBehind in C#'s Page_load():
keyOfVideo.Text = "SomeKey";