If a person forgot to deactivate the light in the refrigerator before Shabbat, under what circumstances, if any, may he open the refrigerator door on Shabbat?

Although Halacha forbids opening the refrigerator door on Shabbat in such a case, it is permissible to ask a gentile to open the door. Opening the refrigerator door when the light will turn on falls under the Halachic category of "Pesik Resheh" – an inherently permissible action that is forbidden because it will inevitably result in a Shabbat violation. In this case, opening the refrigerator door is inherently permissible, but because it will inevitably cause the light to turn on, it is forbidden due to the Halacha of "Pesik Resheh." The concept of "Pesik Resheh" does not apply to gentiles, and therefore a Jew may ask a gentile to open the refrigerator door on Shabbat even though the light will then be activated. (Yalkut Yosef, Helek 4, Shabbat 5, page 229, S"K 8.)

If the refrigerator door was opened on Shabbat and one realized that the light had not been deactivated, under what circumstances may he then close the door?

As with regard to opening the door, Halacha would permit asking a gentile to close the refrigerator door in such a case. Furthermore, Hacham Ovadia Yosef, in the tenth volume of his Yabia Omer, O"H, siman 28, rules that one may close the door with a "Shinui," meaning, in an unusual manner, such as with one's foot or body. He explains that turning off a light on Shabbat is forbidden only by force of Rabbinic enactment, as the Torah prohibition of extinguishing applies only when one extinguishes a flame in order to retrieve the coal. Performing a forbidden activity with a "Shinui" also constitutes a Rabbinic – rather than Biblical – prohibition, such that closing the refrigerator door with a "Shinui" involves two Rabbinic prohibitions. Given that, as mentioned earlier, we deal here with a situation of "Pesik Resheh," rather than a direct act of violation, we may allow closing the door in an unusual fashion. Similarly, it would be permissible to ask a 3 or 4-year old child to close the refrigerator door in such a case. Needless to say, it would then be forbidden to open the door again, unless one asks a gentile, as discussed earlier.

Summary: If one forgot to deactivate the refrigerator light before Shabbat, he may not open the refrigerator door himself, but he may ask a gentile to do so. Once the refrigerator door is open, one may ask a gentile or young child (below age 5) to close the door, or close the door himself in an unusual fashion, such as with one's foot or body.