Reason
R’ Yochanan ben Zakai forbade going up for Birkas Kohanim with shoes or sandals on (גמ' ר"ה דף ל"א ע"ב, גמ' סוטה דף מ' ע"א). The Gemara (גמ' סוטה) gives a few reasons for this.
Honor for the tzibbur. Originally, the Gemara explains that the reason is for the honor of the tzibbur, as people wear their shoes outside and they might be muddy (גמ' סוטה שם ). The poskim explain that we are concerned for this reason too (מ"ב סקי"ח).
Not to spread false rumors. Another reason the Gemara gives is to prevent rumors that a person is a posul Kohen. Shoes often have straps, and there is a concern a Kohen’s shoe strap will break, and while he is sitting to fix and tie it, his fellow Kohanim will go up for Birkas Kohanim, leaving him alone fixing his shoes. People who see this will say the reason he is not going up is because he is a posul Kohen (גמ' שם, רש"י שם, מ"ב סקט"ו).
A Kohen may not go up with shoes even if they do not have straps, as Chazal did not make exceptions to this halacha (ב"י, מ"ב שם ).
Specific Halachos
Completely barefoot. A Kohen should not go up completely barefoot. That would be disrespectful, as people do not appear barefoot before honorable or important individuals (שו"ת הרדב"ז ח"ו סי' רס"ד, וע"פ הגמ' חגיגה דף י"ג ע"ב לאו אורח ארעא וכו', מ"ב סק ח"י ). Even in hot climates, where some people do not wear socks, people still do not usually stand barefoot before an important person barefoot. Nevertheless, a Kohen without socks should not refrain from doing Birkas Kohanim (שו"ת אג"מ או"ח ח"ב סי' ל"ב).
Socks. Although we find that someone with a blemish on his foot should not go up for Birkas Kohanim because people might stare at him and not concentrate on Birkas Kohanim (שו"ע ס"ל ), there is no concern for distraction when it comes to socks. Thus, a Kohen with a hole in his sock does not need to refrain from going up unless it is a big and very prominent hole (מו"ר בשו"ת שבט הקהתי ח"ה סי' רס"ז ).
Similarly, a Kohen may go up with all sorts of colors or patterns on his socks, e.g., “Birkas Kohanim socks,” which have the form of a Kohen’s hands during Birkas Kohanim woven into the fabric, or the like. This is because there are people who wear all sorts of colorful socks all the time, so it does not arouse much surprise (שו"ת אג"מ שם ).
When to remove the shoes. If a Kohen can remove his shoes without touching them with his hands at all, it is best to untie the laces before washing his hands (פמ"ג א"א סק"ט) and remove the shoes after washing, right before he goes up for Birkas Kohanim (מ"ב סקט"ו). However, if he might touch his shoes while removing them, he should remove them before washing so that he does not get his hands unclean after washing them (ע"פ המבואר בשו"ע סי' ד' סי"ח, מג"א סי' קכ"ח סקכ"ו).
Where to leave the shoes. Today, it is accepted to remove the shoes in shul before going up for Birkas Kohanim. However, it is not proper to leave shoes out in the open during the bracha. Instead, they should be placed under the benches out of respect for the tzibbur. This is because shoes are generally dirty or dusty from the street, in line with the reason mentioned above (18).
Right shoe first. Some say Kohanim should first remove the right shoe before Birkas Kohanim and then the left. Since removing the shoes is for a mitzva, the mitzva should be done with the right shoe first (לוח א"י ר"ה, הגר"ח קנייבסקי, דעת נוטה עמ' ס"ד, עי' גליון 97 אות י"ב שהבאנו את זה ). However, others argue and hold one should not exchange the regular practice of removing the left shoe before the right, as removing the shoes is not a mitzva action to prioritize the right side; there is just a halacha of Birkas Kohanim that it must be done without shoes (סופרים וספרים על לוח א"י הנ"ל ). This approach makes sense.
