Here are the steps of how you could save a photo or video from an MMS message on iPhone.
When you have received an MMS message that contain a photo or a video and if you like to
save it down to your device, then here's how:
1. Open up the Messaging app.
2. Then from the conversation, tap on the image.
3. Next tap on the right arrow burst out of the box.
4. Then tap on Save Image or Save Video.
If you cannot see the right arrow appear, then ensure that the controls appears by tapping on the screen.