What would result from not adding fat to pastry dough?

You'd end up with something somewhere between unleavened bread, pasta & laminate flooring, depending on what else you did with it.

The first two are what you'd get if that's how you treated them, the last is what you'd get if you thought you were going to get shortcrust pastry ;)

Late edit
This started out as one of my more flippant short answers, but has since hit the HNQ* list so naturally falls under more scrutiny.
The point of the 'flooring' aspect is that simple flour & water can be used as anything from wallpaper paste raw, or heated & dried as 'hard tack' which really is hard enough to break your teeth.
Pasta can also be made this way, you can omit both egg & oil, then dry it. Try eating dried pasta without re-hydrating.

Of course, such as chapattis are also made simply with flour & water (a splash of oil is optional) This, rolled & dry-heated is of course, absolutely delicious - so it's the treatment after mixing that determines the end product… & its actual edibility.

What you will never get from omitting all fat is pastry… & we're back round to my flooring joke.

*Hot network questions - in the side bar on the right.

From Comments (Thank you PLL) - Glen Baxter said it best (even if he had a different flour in mind)

This sort of pastry is known as short pastry, although it has variants (e.g. you can make flaky piecrust with these ingredients). It is mandatory that you make it with some form of solid fat, the only leeway you have is that you can use lard or butter instead of shortening, other fats such as coconut oil may already require advanced techniques. You also have to use the amount of fat stated, you cannot reduce it.

If you try omitting the fat from this type of recipe, you will end up with something that is nothing like the intended pastry, and it will almost certainly be inedible. Possible results include separate crumbs, a single tooth-breakingly hard crust, or stodgy balls with hard crust on the outside and unbaked dough on the inside, depending on the exact recipe you try to 'adapt'.

Your title mentions oil but your question mentions shortening. Be aware that oil and shortening are not the same thing, and substituting one for the other will change the results. Oil is fat that is liquid at room temperature; shortening is fat that is solid at room temperature (but usually not butter; if a recipe wants butter it will call for it by name). Some pastry recipes are made with shortening and others are made with oil.

If you substitute oil for shortening, you get some kind of pastry, but it's very different. First of all, a 1:1 substitution may not work well. Instead, follow a pastry recipe that uses oil instead of shortening (this one, for example). Oil-based pastries are usually crumblier than shortening-based, and sometimes have an oily mouthfeel. Imagine a slightly greasy shortbread cookie without the butter flavor. Shortening-based pastry can be flaky if done properly; oil-based pastry is never flaky. (At least not in my experience; if anyone has a recipe for flaky oil-based pastry please share.) Oil-based crusts are difficult to roll out, so the "pat-in-the-pan" shaping method is usually used.

Since American Thanksgiving is tomorrow, I imagine you may be making a pumpkin, sweet potato, pecan or apple pie.

• An oil-based crust would work fine for a pumpkin or sweet potato pie. Pre-bake the crust before adding the filling, then adding foil or pie crust shields to protect the edge of the crust from burning. In my experience oil-based crusts have a strong tendency to melt and slide down into the pan during pre-baking. If you use foil or parchment paper to line them during pre-baking, the crust may stick to the foil and rip when you pull it off. A compromise is to carefully cover the edges and top with foil, but leave the base uncovered. Some sources recommend freezing a crust before baking it; I haven't found that helpful but your mileage may vary.

• An oil-based crust would not work very well for an apple pie. Apple pies are usually quite juicy, and the crust will crumble and disappear into the filling, giving you something more like a crumble than a pie. If you must use oil crust for an apple pie, you'll get the best results with a pre-cooked pie filling, so the fruit juices are already reduced into a syrup. Use a streusel or crumble topping instead of trying to shape an oil crust into a top crust.

• I haven't tried an oil crust for pecan pie. My experience with pecan pie is the crust usually burns, and oil crusts have a stronger tendency to burn than shortening-based crusts. So I expect the crust would end up unpalatably burnt. It might be okay if you use pie crust shields for the entire bake time. If you find yourself having to choose between raw filling and burned crust, go for the burned crust. You can always eat the filling by itself and leaved the burned crust on the plate, but you can't (or at least shouldn't) eat undercooked pie filling if the filling has eggs in it.

What you’re proposing to make is the start of what’s known as ‘hard tack’ although it usually has some salt in it, too

It’s a way to make flour shelf stable for a longer period so it could be used as army or ship rations. The dough was rolled out, docked, and then cooked in.a low oven and then left out to dry thoroughly so it couldn’t mold

To eat it, you would have to pour some liquid over it and let it soak for a while before eating or you might risk chipping a tooth

If you don’t cook it as long or at a hotter temperature, it would likely be more similar to a cooking technique from ancient times when ‘pie’ was something where you did NOT eat the crust. It was more like canning— you cooked the food in the crust, then you could hold the food for a few days unrefrigerated as the crust protected the contents. (But distribution of germs and bacteria were different back then, and people’s immune systems would have adapted to it). The ‘coffin’ (crust) would also hold in moisture and prevent the food from burning in a hot oven