Is there a problem opening a refrigerator on Shabbat?
You need to arrange before Shabbat that the light won't turn on when the door opens; then there's no problem
Source: Shulchan Aruch OC 310
In detail
Opening a refrigerator on Shabbat is permitted as long as the interior light has been disabled beforehand. Since turning on a light is a form of 'lighting fire' (one of the 39 melachot), opening the door would indirectly cause a prohibited action. Before Shabbat, one should either unscrew the bulb, use the fridge's Shabbat mode, or tape the light switch. The motor running is not a concern since it operates on a thermostat cycle regardless.
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