I may be wrong but monsters cant close/open doors unless scenario says so. Monsters can attack/move + use special ability.
Monsters do normally have access to the "Open/Close Door" action, which they can use just like heroes do. They auto-fail ability tests if they don't have Attributes, though, which means they can't pick doors like the ones in this quest.
To answer the OP's question, special quest rules override the normal rules. If the special quest rules say you need to pass an attribute test to open the door, and they DON'T say that this test only needs to be performed once, then I would be inclined to say the test needs to be passed each time the door is opened.
Also keep an eye out for anything that might suggest the door cannot be closed again after being picked. But at the end of the day, whatever the quest guide says is what goes.