I was recently in a care home on shabbos and the only way to exit was through a door which could only open if someone pushed a button.
Is it permissible to request a goy to open the door it should one hint? Or ideally wait for a non-jew to use the door?
There are two issues regarding using a non-Jew o do melacha for us on Shabbos or Yom Tov, the prohibition not to ask him to do what we ourselves may not do, and secondly the prohibition not to benefit from the melacha that a non-Jew did for a Jew on Shabbos. Therefore even if you will only hint it to the non-Jew it is will still be an issue of benefiting from what the non-Jew did. Therefore the only viable option is to wait for a non-Jew to open the door, because he wants to walk through it.
Shulchan Aruch O”CH 276-1.