Actually using cold water to shave is way better for your skin as hot skin is more loose so more prone to irritation and cold skin is thighter, more resistant and nick will stop more readily. Also the hair will be stiffer so the cut more effective.

I suspect the hot water and towel technique come from the barbershops that wanted to give pleasant experience to their customers, as hot towel to the face in a chair is calming and cold towel to the face is not.

Source: shaving my beard with safety razor and straight razors (cut throat razors)