The gemara says that one should first remove his left shoe and only then his right shoe [in order to give honor to the right - Rashi].
One Talmid Chochom suggested that when removing one's shoes before Yom Kippur he should first remove his right shoe and then his left shoe. The reason is that since it is a mitzva to remove his shoes before Yom Kippur, he should start with the more choshuv right shoe.
This is not compelling. Removing one's shoes before Yom Kippur is not a mitzva for if he would not be wearing shoes in the first place he would not have to put them on and then take them off. This proves that it's not a mitzva.
In addition - we remove our shoes as affliction. We aren't concerned with chashivus in the context of affliction. Thus, we should still remove our left shoe first as always [R' Shlomo Zalman ztz"l].
ספר דף על דף עמ' רצה