Does touching new shoes out of the box (never worn) require one to wash his hands?
There is no need to wash hands after touching new (never worn) shoes.
The concept of washing after touching shoes is either on account of dirt, or on account of a ru’ach ra’ah of shoes (which some authorities write doesn’t apply today).
The Shulchan Aruch (Orach Chaim 4:18) writes that the reason for washing hands is because of dirt (see Mishnah Berurah 41), in which case the halachah won’t apply for new shoes. This is explicit in the Kaf Ha-Chaim (71).
The Magen Avraham (128:26) notes the reason of ru’ach ra’ah, and based on this apply the principle even to clean shoes (see also Shulchan Aruch Ha-Rav 128:27).
However, the Ya’avatz (in Mor U-ketziah) writes that even according to this possibility, the ru’ach ra’ah that descends upon shoes only applies to shoes that are worn, and not to new and never worn shoes. Therefore, there is certainly no need to wash after touching new (never worn) shoes.