Why doesn't Robert Boyle's perpetual flask work?

It does not because loss is too high. If you replace the loss by adding a pump, it will work. It is similar to how oscillators work. The flask is similar to a swinging pendulum. read more

By Robert Boyle’ perpetual flask, I assume that you mean a container with a drain and the pipe from the drain empties back into the container. It does not work, because the end of the pipe is higher than the fluid level in the container. read more

Boyle's self-flowing flask, a perpetual motion machine, appears to fill itself through siphon action ("hydrostatic perpetual motion") and involves the "hydrostatic paradox"[17] This is not possible in reality; a siphon requires its "output" to be lower than the "input". read more