No, that fact that the address on your driver's license is not current will not delay your tax refund.
Your driver's license number is actually not required to file your federal taxes, nor is it required to file your State tax return in most states. If your State issued ID is not being accepted, you can leave the driver's license/state ID information blank, and still file your federal tax return.
Two states, Illinois and Alabama, do require a driver's license to file a state return.
For additional information, you can view the IRS information on New Security Safeguards. The two following paragraphs taken from that website are specific to your situation:
Do I need a driver’s license number to file a return?
You do not need a driver’s license number to file a federal return. 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. This is one more layer of protection against identity thieves. State tax software will prompt you for your driver’s license number if it is requested by the state. You also can review state revenue department websites for information.
Why am I being asked for my driver’s license number?
In an effort to better protect you from identity thieves, some states will be trying new approaches. Some states may ask for additional identification information, such as your driver’s license number, when you are preparing your state tax return. This will be another layer of protection because identity thieves may already have your name and Social Security number, but perhaps not your driver’s license number. States requesting this information have the ability to match driver’s license information and other identifying records to help confirm your identity. You do not need a driver’s license number to file your federal tax return.