Can you leave the water running over night with a portable dishwasher?

You can, however if your faucet can't handle the pressure having nowhere to go that long then you might have a chance of damaging it. However the pipes, the underworkings of the sink, etc. Will be fine. I'd advise against it if you have one of those side hoses for spraying things as the hose might not like prolonged exposure. But once in a while won't hurt anything. Like ware and tear, stress has to build first. Just don't make it a habit.