A closed/unpowered iron door is unaffected as far as i can tell, but if the door is powered and open a villager can use it as if it was a wooden door and open and close it. I also only tested this on double iron doors but both doors are still affected. both doors are connected to the same lever. in the attached images a baby villager walk in one door close it and out the other and close it, without the switched being moved.
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I can confirm, quite annoying, Villagers can close (don't think they can open) powered iron doors. My villages ALL have iron doors powered by underground redstone connected to a single lever which I can use to close all doors when a raid start. Now the villagers will start closing the doors and lock themselves in, in a few minutes.
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We're already tracking this issue at MCPE-44621, so I will resolve and link this ticket as a duplicate.
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I also found that wandering traders can open and close iron doors. I know that wandering traders will try to keep a distance between zombies and other enemies, so if there is a door in their way, they will open it and then shut it, and this includes iron doors.