I can read that changing the app icon programmatically at runtime is not possible, at least since iOS 3.0.
But... I regulary can see on "App" websites apps that can put icons on the springboard. The first I remember is that kind of app that you give a friend's picture from your album, and it creates an icon so you can call him just clicking on that icon:
http://itunes.apple.com/fr/app/speedtouch-the-best-home-screen/id378360978?mt=8
http://itunes.apple.com/fr/app/desktop-shortcut/id421701004?mt=8
http://itunes.apple.com/fr/app/appbutler-app-organizer/id327391626?mt=8
http://itunes.apple.com/fr/app/tap4music/id365578914?mt=8
So I guess, is it still impossible to change the app icon programmatically at runtime?
Apple have changed this, you can do this now: https://developer.apple.com/documentation/uikit/uiapplication/2806818-setalternateiconname
What you're thinking of here are WebClips, which are basically shortcuts to websites on the home screen. An actual application cannot change it's icon at runtime, as it's defined in the static Info.plist file.
The only exception to this rule is Newsstand applications.