If your return is still in progress, you can change the zip code by visiting the Personal Info section of TurboTax Free Edition online. Scroll down to the mailing address section, click edit and change the zip code information.
If your return has been efiled and accepted, you are unable to make any changes because the return is already being processed. You may want to call the IRS and inform them of the zip code change in a couple weeks once your return is fully processed by filing IRS form 8822. The IRS contact phone number is 1-800-829-1040.
If you move before filing your tax return, put the new address on your return. The IRS will update your records when they process your return.
If you move after filing your return, you should have the post office forward mail to your new address. Some post offices do not forward government checks, so you may also want to change your address with the IRS. You can use IRS Form 8822, Change of Address to notify them of an address change.