What is the literature on long term water fasts?

Up front: I am biased against extreme diets like water-only fasts. I can see a use case in carefully medically supervised settings, such as for a cancer treatment, and I know that some religious practitioners use them. I've never tried them and have never been morbidly obese.

The only truly releveant paper I found that looked relevant was a case study of a woman whose 40-day water fast caused thiamine deficiency, which led to her developing a severe neurological disorder called Wernicke's encephalopathy (source).

The academic literature on prolonged water-only fasts is extremely limited, particularly for weight-loss purposes in humans. Prolonged water-only fasts have a few more studies in non-human organisms and are sometimes used as anti-cancer treatments.

I would be really careful about basing decisions on reddit anecdotes. Questions I would be wondering about:

• What nutritional deficiencies and other potentially serious health risks might result from a sustained water-only fast?
• For those who lost weight on a prolonged water fast, did they keep it off long-term?
• Of those who try a water fast, how many fail to sustain it?
• How much selection bias in favor of success stories when people post their weight-loss experiences in /r/fasting?
• How does a failed extreme dieting attempt affect your motivation to try again in the future?

The metis in the dieting world appears to be that crash diets tend to be yo-yo diets, leading either to failure of the diet attempt or to rapid weight regain after the diet ends. My picture of successful dieting tends to look more like what I'm doing:

• Eating a lower but sustainable number of calories per day (like 1200 calories for a 200-pound person), mainly by replacing oils/fats, sugars and starches with fruits and non-starch vegetables and by improving the ability to eat limited portions
• Exercising, especially resistance training, which has the best evidence for counteracting the decrease in resting metabolic rate that occurs when you start dieting
• Gaining intellectual knowledge of nutritional concepts like resting metabolic rate, caloric density, glycemic index, and how the shift from water/glycogen-burning to fat-burning affects your rate of weight loss 
• Developing a typology of your hunger symptoms, what they seem to mean (i.e. my stomach isn't full vs. my blood sugar is low), whether they should motivate you to eat, and if so, what type of food and how much. For example, in my case, stomach rumbles and a feeling of emptiness seems to mean my stomach is empty, whereas wooziness, shortness of breath, and irritability seem to mean my blood sugar is low.
• Figuring out at what stage of hunger and in what circumstances you do or do not have self-control over your eating, and making sure to avoid circumstances where you lose control and impulse-eat
• Adjusting your shopping, meal-planning, and restaurant-ordering behaviors
• Getting socially connected: talking about your goals with family and friends, with a doctor or nutritionist, with an exercise coach, and being at least occasionally involved with support gorups
• Making it fun and motivating: building up a positive picture of what benefits you expect to reap and a clear-eyed understanding of where your weight and weight-linked physiological traits are now and where you're headed if you don't change course
• There are amazing reports about the new weight loss drugs like Wegovy, but I have not tried them and they are currently very expensive and hard to get

Please don't take this as medical advice - it's just a description of what I'm doing. I'd highly recommend exploring a range of dieting options and considering each on their merits before selecting one, and ideally interfacing with a medical professional to consider your specific situation.

Non professional opinions of a 60kg guy who was full-time obsessed with learning nutrition/fasting for 2 months.

Literature on long term water fasts is severely lacking, most likely because it's not profitable to promote anti-consumption in consumption society. Reading medical studies directly is a great way to understand fasting and nutrition. With better understanding of nutrition it's much easier to see junk food for what it is. And in consequence it loses its appeal.

1. Prolonged water fast should not have negative impact on future metabolism. It is highly advisable to consult medical professional before and during prolonged fast. They can help monitor concentrations of vitamins, minerals and expelled protein. Unmonitored prolonged fast can cause deficiencies and those can cause severe medical problems. Non-adults, pregnant individuals or those suffering medical conditions that can be negatively affected should not attempt prolonged fast. Most common adverse effects include: bad breath, mood swings and deregulation of menstrual cycles. Anecdotal effect which seems more important than loss of fat is shift in worldview, proponents of prolonged fasting include Moses, Buddha, Jesus, Plato, Pythagoras and Gandhi.
2. After initial period during which body uses mostly stored carbohydrates and protein for energy, you can expect to lose 200g-300g of fat per day. To shorten the initial period, lower its adverse effects and "ease into" prolonged fast it is beneficial to eat only one heavy in fat meal per day for two weeks in advance.
3. Before the fast it's important to have adequate levels of sodium and potassium. They are expelled in normal quantities at the start of prolonged fast and it takes body some time to stop it. To mitigate effects similar to those of dehydration during first few days you can eat teaspoon of salt per day and progressively lower it. To mitigate hunger which will be most intense during first few days, you can "cheat" and drink tea or coffee (pure). After the initial period, hunger should subside, anecdotally it will return on its own after undetermined amount of time. When it returns it is very important to break the fast gradually. Start with something like a glass of broth, next day something equivalent to one egg, then two and four. Prolonged fasting can help reduce weight in long term, it can assist in changing eating habits. Best motivation to eat healthy is knowledge of nutrition, weight is just a byproduct of it. 

Here I cite reddit posts, not literature, because /r/fasting has a lot of good anecdotal data, and many weight loss studies are limited in scope. 

The answers to any of these questions will likely depend on your starting weight. 

On Question 2: In theory this is just a function of your BMR (basal metabolic rate) and TDEE (total daily energy expenditure). For example, if you are large enough to have a TDEE of 3000kcal, then you will lose 1lb of body mass per day (how much is muscle vs fat unclear). 

In practice this is a bit of an overestimate. For anecdotal success stories you could go to /r/fasting. On Top All I see:

• 104lbs lost in "4 months" (14 day fasts followed by 5 days keto at a slight deficit) = 104lbs in 90ish days of fasting  (morbidly obese man)
• 90lbs lost in a 40 day water fast (morbidly obese man)
• Many (often smaller) people doing OMAD, rolling 48, etc

Searching for "14 day" I see: (keep in mind, about 10+lbs of this is water weight)

• 23lbs in 14 days
• 30lbs in 14 days
• 25lbs in 14 days

Common wisdom on this subreddit is you get 0.5lbs/day of "real fat loss" during an extended fast. 

For point one , yes. We have evidence that your body has a steady state homeostatic "weight" that it will attempt to return you to. Which is why on the whole all fad diets are equivalent and none are reccomended.

"Non metabolic" is sort of a vague statement but top of my head besides "organs" i'd imagine the possible gut flora problems could be huge (or it might be great because presumably you have flora right now encouraging excess fat etc)