Twitter has currently no option to save a GIF directly in their app. Another problem is that Twitter GIFs are not real GIFs. They are just videos that automatically play in a loop. However, it is possible to download a Twitter GIF and convert it to a real GIF in just two steps.
First, you need to download the GIF as video. SaveDeo makes that possible:
Now you have the GIF in the MP4 video format. Next you need to convert the video to a GIF. Therefore you can useEZGIF: