Must one wash his hands if he removed his shoe without using his hands?<a href="#_ftn1" name="_ftnref1">[1]</a>
No.
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<a href="#_ftnref1" name="_ftn1">[1]</a> Admur 4:18 “hand”; M”A 4:19; Shiyurei Kneses Hagedola 128:4; Elya Raba 4:13; Mamar Mordechai 4:9; Chayeh Adam 2:5; Kaf Hachaim 4:69; Nimukeiy Orach Chaim 4:3
Other Opinions: Olas Tamid 4:12 [brought in Peri Megadim 4 A”A 19; Ashel Avraham Butchach] rules one is required to wash his hands after removing the shoes even if he did not use his hands. We do not rule like this opinion.

Shulchan Aruch HaRav | ❓Must one wash his hands if he removed his shoe without using his hands?

Must one wash his hands if he removed his shoe without using his hands?[1] No. [1] Admur 4:18 “hand”; M”A 4:19; Shiyurei Kneses Hagedola 128:4; Elya Raba 4:13; Mamar Mordechai 4:9; Chayeh Adam 2:5; Kaf Hachaim 4:69; Nimukeiy Orach Chaim 4:3 Other Opinions: Olas Tamid 4:12