Many skin conditions can affect the feet and many systemic skin conditions that affect the whole body can be worse in the feet due to the use if shoes and especially the loads that runners put on their feet.
Blisters
Blisters are common on the feet of runners. They are due to superficial friction from the running shoes on specific areas which leads to the skin layers being separated. This can typically be on the toes or the back of the heel. Most runners are very familiar with blisters. The best way to deal with blisters is to not get them in the first place. Make sure the running shoes fit properly and use a lubricant on area of high friction. There are numerous friction relieving tapes or pads that runners can use to prevent blisters if the occur frequently. If a blister does occur, then make sure it is protected and padded to allow it to heal up.
Corns and Callus
Corns and calluses are caused by to much pressure on an area. This typically is due to the shoes not fitting properly or there is some deformity causing increased pressure such as a hammer toe or bunion. Corns and calluses are relatively easy to get rid of. A skillful podiatrist can easily remove them with debridement. What is not so easy is to stop them coming back. Corns do not have roots that they grow back from that the podiatrist did not remove. It is a common myth that corns have roots. Corns and calluses always come back if the cause is not removed and that cause is the excessive pressure on the area. That pressure could be relieved by, for example, surgery to correct a hammer toe or bunions. Foot orthotics will help relive pressure on the bottom of the foot if there are calluses there.