Would there any damage done to an engine if you run it without a thermostat permanently? Let's assume the vehicle is driven in a climate that is warm and temperatures never drop below freezing.
Engines are designed to run within a specific temperature range.
The components of the engine expand as the engine warms up. The designed clearances within the engine will be correct around the temperature that the thermostat maintains.
If no thermostat is installed, the engine will take longer to reach the optimum designed temperature and probably wont ever reach it if the rest of the cooling system is operating efficiently.
With the engine running cooler than designed for, there may well be increased wear due to the non-optimal clearances between the moving parts.
Most of the wear in the engine occurs while cold. Only when the operating temperature is reached, the metal part clearances are where they should be and oil is at the correct viscosity.
If you take off the thermostat, your coolant will always flow in the "big loop", which will delay engine warmup significantly and cause a lot more wear over time.