I observed the following on my iPhone:
1. When connected to the local LAN via wifi, I can navigate to the device page for any device using HTTP and get the device status, but the change status buttons ("On", Off", "Dim", etc.) do not work. The HTTP response times out.
2. HTTP interface also does not work via an Internet connection (with port 80 open on the firewall). But who wants to open their private network to insecure traffic anyway?
3. HTTPS interface works fine when iPhone is connected to the LAN via wifi.
4. HTTPS interface does not work over the Internet until a self signed SSL certificate is installed on the ISY. With the default certificate, iPhone gives the user an "invalid certificate" warning. Even if you click the Continue button to ignore the warning, iPhone will give a "network connection lost" error. If you install a valid self-signed certificate, the warning goes away and the interface works perfectly.
Since the HTML UI is so lightweight anyway, I don't notice any performance difference between connecting over the LAN or Internet, so I just bookmarked the public address and access my ISY from there, whether I'm in my house or out.
-Jim