Can I wear cuff links with a shirt that had buttons?

Typically, no, since there would be no facing (bottom side) hole or holes to receive the link fastening part. A good seamstress could, however, remove the button and create a buttonhole in its place if (and only if) the cuff were long enough to accommodate the shirt wearer's wrist with the cuff ends overlapped for insertion of the cuff link. For the typical shirt wearer of casual, not formal, dress wearer, the more practical solution for lost buttons would be to obtain and sew on replacement buttons as needed, or to purchase a new shirt.