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R Howard Jachter wrote here a long article on the topic -- find a summary below with light edits. To immediately answer the question of those who rule more strictly he cites R Tzvi Pesach Frank, R Mordechai Yaakov Breish & R Yitzchak Yaakov Weiss and those who recommend that one to be strict if possible: Rav Yosef Henkin and Rav Ovadia Yosef.

Rav Shlomo Zalman Auerbach rules that it is entirely permissible to open a fridge door when the motor is not running. In fact, he writes that it is not right to be strict on this matter, as it will lead one to limit his Oneg Shabbat, enjoyment of Shabbat.

His reasoning is based on the fact that opening up the door will not immediately lead to turning on the motor. The inevitable time delay between the opening of the door and causing the motor to go on leads Rav Shlomo Zalman to classify this as a "Grama" - "Koach Sheini" (indirect action).

Now the Rama (334:22 and see Biur Halacha ad. locum. s.v. D'Gram Kibui) rules that Grama is permitted only in situations of great need. Rav Shlomo Zalman asserts, however, that since one's intention is merely to open the door and not to turn on the refrigerator's motor, Grama would be permissible in all situations even absent any unconventional needs. Moreover, he writes that since he is only causing the motor to go on earlier than it would have gone on without his opening the refrigerator door, (also see Minchat Shlomo 91:10) one may treat the act of opening of the door even more leniently than Grama. Thus, opening the refrigerator door would be permissible in all situations.

Many poskim concur with Rav Shlomo Zalman's lenient approach. Indeed, Rav Aharon Lichtenstein told this author that Rav Soloveitchik agreed with the lenient approach. Rav Moshe Feinstein (Igrot Moshe O.C IV: 74- Bishul -28) seems to fully accept Rav Shlomo Zalman's ruling (also see Igrot Moshe O.C. 2:68)- as does Rav Eliezer Waldenburg (Tzitz Eliezer 8:12 and 12:92). (See Encyclopedia Talmudit 18:663 note 13.)

Many eminent authorities, on the other hand, either rule strictly (Har Zvi O.C. I:151; Chelkat Yaakov 3:179; and Minchat Yitzchak 2:16) or at the least recommend that one to be strict if possible (Rav Yosef Henkin, Eidut L'Yisrael p. 122; Rav Ovadia Yosef, Yabia Omer I:O.C. 27). The problem is that once an action is performed routinely it cannot be classified merely as Grama (see Shabbat 120b and Rabbeinu Chananeil ad. loc. s.v. Rav Ashi and Bava Kama 60a and Rosh Bava Kama 6:11). Rav Shlomo Zalman responds that this applies only when one intends to create the resultant action. When opening the door one does not intend to turn on the motor.

R Yair Hoffman writes here that the answer changes over time as refrigerator technology evolves and brings additional views of whose who disagreed with RSZA

  • Rav Eliyahu Henkin, zt’l, ruled that one can only open the refrigerator when the compressor is on and it is forbidden to do so when the compressor is off
  • This was also the view of Rav Elyashiv, zt’l, as cited in Mitbach K’halacha, page 303
  • The Chazon Ish ruled that it is forbidden to open the refrigerator at all, whether the compressor is on or off. (Cited in HaPardes, Cheshvan 5719/1958)

He writes that issues are actually becoming more complicated with modern refrigerators and brings additional poskim being concerned

  • Rav Hershel Schachter holds that recording the information on a computer [as is done in modern fridges] would under many circumstances at least be considered ksiva miderabanan. He also feels that there may be issues of davar shemiskaven–the person intends for it to happen since it is beneficial for him–even on a rabbinic violation.

  • Rabbi Eli Gersten of the OU stated that Rav Yisroel Belsky pushed to have timers on refrigerators. He told Rabbi Gersten that his grandfather, Rav Wilhelm, used to sell timers in his hardware store–they called it “the Rav Henkin switch.” Rabbi Gersten also stated that Rav Belsky used the language of “chashashos,” concerns, of a Torah prohibition. Rabbi Gersten further stated that when one student from Yeshiva Torah Vodaath (who accompanied Rav Belsky at the OU on Thursdays when he came to rule on questions) expressed grave concerns after purchasing a new refrigerator, he said, “Look, until a timer comes out, just open it with a shinui. When the timer comes out, purchase it.”

See also here from Star-K and here for additional concerns.