On an Android phone, a search in the Google Play Store for "Live Wallpaper" yields a large number of animated and even interactive wallpapers. On iOS the selection is generally limited to those animated backgrounds that shipped with your phone, though there are ways around this limitation.
Windows Phone does not use wallpapers in the same way as the other two major phone operating systems, but by using Microsoft's "Live Lock Screen" app, a user can deploy animated and interactive backgrounds for the phone in its locked mode.
Although the iPhone selection of wallpapers is limited, users with so-called "jailbroken" phones can install animated gifs as their desktop background by using a variety of sideloaded apps dedicated to that task. However, remember that apps obtained from outside of the official Apple App Store have not been reviewed and approved by Apple, which means users run the risk of malware or simply poor functionality.
There are also many Live Wallpaper apps for Android that allow users to deploy their own animated gifs on their Android desktop, but part of the joy of Android Live Wallpaper is in more dynamic backgrounds that respond to touch, to movement of the phone, or even allow users to play simple games without opening an app.