Most state DMVs do not track citizenship status. Only update your DMV with your new citizenship status if your state tracks that information. Your DL usually contains no information about your citizenship status. I would just wait to renew it as normal if your state doesn't typically note citizenship or alien status on the DL.
The REAL ID Act makes it a requirement for states to issue drivers licenses that verify the lawful status of the applicant. For now all states have been granted an extension to implement the REAL ID Act, but eventually it will become a reality.