Feet are highly complex structures. Foot defects are very common, and a defect affects how stresses placed on the feet are borne by the feet and the rest of the body. When forces on the feet are exaggerated by high-load activities such as heavy weight training, and running, the defects are magnified. While foot defects might not be the whole story behind foot, ankle, knee or back problems, they are contributing factors, perhaps major ones.
It is recommended to wear high quality shoes with molded internal soles, to compensate for any structural or postural instability in the feet, or, in the case of defect-free feet, to maintain that existing good condition over the long term. While orthopedic shoes with custom-made molded internal soles are the ideal, off-the-shelf shoes with molded internal soles are, in most cases, light years ahead of no support. You can get molded internal soles that can be slipped inside your regular footwear (but you may need to remove some of the existing soles to make room).
A change in footwear, or the insertion of high-quality molded soles into your existing footwear, will help greatly to keep you lifting intensively, heavily and free of injuries. It is also highly recommended to consult a podiatrist, preferably a sports podiatrist, or a chiropractor with training in orthotics, to get your feet checked out.