The poles are to remain in the rings of this ark; they are not to be removed.
When the camp is to move, Aaron and his sons are to go in and take down the shielding curtain and cover the ark of the Testimony with it. Then they are to cover this with hides of sea cows, spread a cloth of solid blue over that and put the poles in place.
There are a couple different ways of resolving these passages. One is to consider the rule in Exodus as having common-sense exceptions, just as Jesus observed there were exceptions to not working on the Sabbath (Mt 12:5, 11-12) as well as other laws (Mt 12:3-4). It was important for people to acknowledge God's holiness by not touching the Ark, but it was also important that the laymen not see the Ark and thus be in the direct presence of God. If removing the poles were necessary for the Ark to be securely covered with cloth, better that than the chance of someone seeing the Ark.