The "Always on Display" feature was first launched by Samsung in Galaxy S7 & S7 Edge since Feb, 2016.
Samsung has now improved its "Always On Display" feature which now have the option of show GIF's images as well with the help of updated AOD software. Before that it only display time, notification, battery info or a still image when the phone is idle.
However there is a restriction on the size of GIF, which you need to shorten it by going to the edit button in the gallery. Also the GIF plays once when you turn of the display. This is all done in order to make GIFs battery efficient.
Currently this feature is only available in Android Oreo with the latest update of "Always on Display" i.e. version 3.2.26.4. and it's working on Samsung Galaxy S8, Galaxy S8 Plus & Note 8 and expected in the Samsung Galaxy S9 & S9 Plus.
You can download the latest version of "Always On Display" from here.