Open the GIF in Safari, Messages, Mail, or another app
Tap and hold the GIF
Tap Save Image or Add to Photos
Open the Photos app
Go to Albums
Open Animated or Recents to find the GIF
If needed, use the Files app to save the GIF from a download link
If the GIF is in Messages, tap and hold it, then tap Save
If the GIF is in Safari, tap and hold the GIF, then tap Add to Photos
If the GIF does not animate in Photos, check that it saved as a GIF file and not as a still image