Thank you for your question. If you received a citation or were arrested for DUI, you might be in the temporary license period (30 days), which is typically extended if you request a DMV admin per se hearing.
Before the DMV takes any action against your license, absent other factors, you have a full license. That means that you can renew it, just as though the DUI had never happened.
However, if the DMV takes action against you, or if you are convicted in court, your license, even if renewed, can be suspended, or revoked (if a refusal or if you have a high number of points).
Your license status with the DMV, as they say, is only as good as their computers say it is. That is more important than whether or not you have the physical license renewed or not.