Do I use zip code on W2 form (old address) or new zip code?

Enter the W-2 exactly as printed. If what you report to the IRS does not match what the issuer of that W-2 reported to the IRS, you will have issues with the IRS. As for your tax return, the only address that matters to the IRS on *your* tax return, is the mailing address you enter under the Personal Info section.

As Carl mentions above, enter whatever is on your printed W-2 even if the address is not current.  The address you are entering is specific to that W-2.  Your current and correct address (entered in 'My Info') won't be affected by what gets entered from a W-2.