[cornedor]

The icon only changes on April fools as a joke or something like that. And it changes for everybody. At least on iPhone, I don't think you can change the app icon without notifying the user.

[Izkata]

Last year in October they changed it to a melting version of the owl to get users' attention: https://www.digitaltrends.com/mobile/why-is-duolingo-melting...

[watwut]

There is widget with owl that gets sadder and sadder as the day progresses if you did not done duolingo.