Depends. There are instances that a whole row of train stations may not be open but I doubt you will go on any of them since they're usually on Brooklyn or Queens-bound trains like the J or L lines. Most of the time, they will just run local which can be a pain in the ass or tell you to take a different train route if there's construction. However, the NYC Subway System is so connected in Manhattan that there's at least a way to get anywhere and Manhattan rarely has a shuttle bus service because Manhattan bus are SOOOO SLOOOOW. It's a test of patience and time management. The NYC Subway website always update every week if there's service interruptions.

It might be much worse at night after 11:30 PM (when night service begins) depending where you are, but the upside they run all day and all night.