The doors operate on a infrared sensor.
'Technically' by walking in front of them, even 'piggybacking' when the infra hits you, it resets the timer for the doors to close. there really is no avoidance to this, especially if there is a fire alarm on the back door, and the windows are barred.
Since there is no way around it, Ultraorthodox will avoid having reservations at such hotels.
Your options would be to not use it at all, or to do so.
Your responsibility would be to ask YOUR rabbi, as there are different views between rabbinic studies and sects of Judaism, or communities.
Since automatic doors were not invented until late, different opinions will vary.
As I've been taught, you should follow the teachings of your local Rav. If he doesn't know, its his responsibility to talk to higher chain in command. If another Rav contradicts him, then you must still follow your Rav's teaching, as diferent optinions will turn to different outcomes. Since we do not have a Sanhendrin anymore, there is no one specific law for this question