You can still use it while expired, for return purposes. That won't affect your return. It's an additional measure to protect your identity but not required, in all states. There are only a couple states that require you to enter it in order to file.
Some states may request your driver’s license number for state tax returns because they have the ability to match state records and help confirm your identity. But if you do not enter it at all, that may slow the processing time of your return.