I am primarily interested in intermittent fasting for health (by promoting autophagy), not weight loss. Paul Jaminet suggests here that one can continue to eat during the fast, so long as protein is restricted and carbohydrates come from fibrous sources. I question whether consuming calories of any type during a fast breaks it. Does autophagy continue if I only eat fat? Are there any studies to confirm or reject this?
There are lots of benefits you get from fasting, some of them you get just from protein/carbohydrate restriction, some of them you don't. Whether or not you call it protein/carb restriction "fast" or not, is a purely verbal matter. There are relevant differences and similarities between eating only fat and not eating anything. People have, however, disagreed about which is the real definition here and here.
As to the actual question about autophagy. You'll still get autophagy from protein restriction alone, since the amino acids have to come from somewhere. Carbohydrate would reduce autophagy I strongly suspect, given that it'll spare protein. Eating adequate carbohydrate but restricting protein, you'd be doing something very unlike a fast (though still valuable). Eating plain cream or some such, would plausibly reduce autophagy in the short term, as it does increase insulin which does inhibit autophagy. I would imagine, given that even thinking about food increases insulin, that pure fat would so the same, but whether it does so enough for you to care is up to you. If eating some fat allows you to extend a fast, then I can only assume you're gaining a net benefit.
The studies that he cites demonstrate that it is glucose and amino acids that cease autophagy, not fat so indeed if you only eat fat you will reap the benefits. Staying in ketosis all of the time begets the same effect but he doesn't think that we should be in ketosis most of the time so he advocates keto-fasting.
I'm not sure about the other benefits of fasting though. There might be others that get negated by any food.