In Firefox for iOS 1.2 you can long press on an image and then select the Save Image menu item. The image will be saved to the Photo Library, which you can access through the standard Photos application.

If the menu does not appear, it is possible that the web site has implemented a protection against image copying.

I don't believe it has anything to do with protection against saving the image because the images will save in Chrome or Safari, but will only sometimes work on Firefox.

For example, DeviantArt has no protection against saving images. I can save images just fine from that site if I'm using Chrome or Safari. (Hitting "Download" on the artist's post page usually loads a higher res version of the image, which I can then long press to save within those browsers.)

However, if I load those same DeviantArt pages in Firefox, hit the download button, then long press the image that loads, it doesn't work. No menu appears at all, no matter how long I hold my finger on the image. On seemingly random occasions it has allowed me to save the image, though.

During the times it wouldn't let me save the image in Firefox, I was able to open the same image and download it in Safari.

I don't know what is going on to enable this behavior. It's certainly strange.

It's possible you did not grand Firefox access to Photos on your iPhone. You will need to grant that permission, which is prompted the first time you attempt to save a picture on to the device.

Testing locally right here, I am able to save pictures from Deviantart to my iPhone just fine.

Firefox had permission to access photos. Otherwise, I wouldn't have been able to save at all.

I redownloaded the app because none what's been happening makes sense. After doing that, it is now letting me save pictures regularly. Perhaps something got borked on the initial install?

Anyway, thanks for trying to help.