Can I e file with a expired id?

Yes, you can still file your income tax return.  If you are having a problem entering your expired ID, leave it blank and continue with your filing.  An ID is not required to E-file.  

I selected No ID, however, the IRS rejected my tax return. Turbo Tax suggests that I should update the ID or mail the tax return.

You will be able to use TurboTax to file even if you don't have your ID. You are not required to have a driver's license or state ID card to e-file your federal taxes through TurboTax. Some states require you to provide that information in order to e-file. If you live in a state that requires ID, you will need to wait until your ID is replaced, or print and mail in your state tax return.