TwiDropper is a free web tool that allows you to easily save videos, GIFs, and images from Twitter on your iPhone.
You can save videos, GIFs, and images directly to Dropbox, an online storage service, thus saving storage space on your device. Of course, you can also download them to your device without using Dropbox.
You can save videos, GIFs, and images of tweets from private accounts. In order to do so, you need to log in with the account you are following for that private account.
You can save videos, GIFs, and images sent and received via direct messages (DMs).