Engine parts are designed to fit properly when they run at a specific temperature. Pistons in particular are oval-shaped when they're cold and expand to fit correctly when warm. Since they don't fit properly when they're cold, that's when most engine wear takes place.
Besides lubricating, engine oil's job is to collect and carry away dirt and blowby. Piston rings don't seal well when they're cold so you'll have a lot more blowby. When the oil is hot, unburned gas fumes vaporize off the oil and are drawn out by the PVC system to be burned in the engine. When the oil is too cold, that blowby collects to form sludge. Now you have more blowby, more sludge, and less chance to get rid of it. Cold oil doesn't lubricate well either. It will be thicker so it doesn't flow as easily. In particular, it squirts out of a tiny hole in the connecting rods and up onto the cylinder walls to lubricate the pistons. The oil film is necessary for helping the two top piston rings to seal. Without that oil on the cylinder walls, the soft aluminum pistons will scuff and start to peel apart.
What it will hurt depends on your definition of "hurt" and the time frame you're looking at. Nothing is going to self destruct and fly apart but 99 percent of internal engine wear takes place while the cold engine is warming up. That's why vehicles that see almost only highway driving reach such high mileages. My '88 Grand Caravan has close to 400,000 miles and that is considered normal by fleet operators. Without ever having a thermostat installed, I doubt any engine would reach 50,000 miles and even that might be dreaming.
On newer engines the cold engine will set diagnostic fault codes and there are some self-tests the Engine Computer won't run. Those tests are what helps it identify and detect problems with the many sensor circuits and operating conditions. When the tests don't run it won't set some fault codes and without those it's almost impossible to know where to start troubleshooting to diagnose a problem. If your vehicle has an oxygen sensor in the exhaust system, the Engine Computer uses its readings to fine tune the fuel / air mixture for best emissions, best fuel mileage, and best performance. That's called "closed loop" operation. The computer doesn't even look at those readings, meaning it stays in "open loop", until it knows the sensor is hot enough, (600 degrees) to be accurate. It knows that by going into closed loop when the coolant temperature sensor reports a certain temperature, typically around 180 degrees. You'll never reach 180 degrees without a thermostat.
Gasoline doesn't vaporize well when it's cold, and liquid gas does not burn. It goes uselessly out the tail pipe. That's why we needed a choke before we had fuel injection. It was to dump more raw fuel into the engine in hopes a high enough percentage would vaporize to make the engine run satisfactorily. For decades we had automatic chokes. They operate on some form of hot exhaust gas or hot coolant, or a combination. There is also typically some passage under the carburetor that warms the base of it to help the gas vaporize faster. When your engine stays too cold, you'll have to put more gas in to get it to run right. A lot of the unburned gas will wash down the cylinder walls past the piston rings. That will wash the oil film off and lead to more blowby and greatly accelerated wear.
In rare cases a lack of a thermostat can cause engine overheating. The hot coolant doesn't stay in the radiator long enough to give up its heat to the surrounding air.
There is no valid reason to leave the thermostat out, and as you can see, there are a lot of issues in doing so. Reduced fuel mileage, reduced power, increased engine wear, and increased emissions are all possibilities. You won't notice the engine wear right away, and it can be hard to tell because it gradually gets worse over time, but it IS noticeable when you drive someone else's identical vehicle that was maintained properly. It can have much higher mileage but will run better.