Schools and colleges will remain shut down/shut/closed tommorrow due to bad weather. Here we can use shut, shut down and closed either one right?Thank you.
In the context of a building, a plant, a facility such as a school, hospital, airport etc, "...will remain shut (without down) is not idiomatically correct. You can say " keep the window shut / closed", meaning do not open it.
Wall Street was shut down for two days by hurricane Sandy. Stock market was closed due to flooding.