As far as I remember, no need to renew the GC. Don't spend money on this. For your peace of mind you could go to your local USCIS office for a passport stamp, this would be your temporary green card.
This is from the guide to naturalization:
24. If my Permanent Resident Card expires while I am applying for
naturalization, do I still need to apply for a new card?
If you apply for naturalization 6 months or more before the expiration date on your
Permanent Resident Card (formerly known as an Alien Registration Card or “Green
Card”), you do not have to apply for a new card. However, you may apply for a renewal
card if you wish by using an “Application to Replace Permanent Resident Card” (Form I-
90) and paying the appropriate fee. Call the USCIS Forms Line or visit
www.uscis.gov.
If you apply for naturalization less than 6 months before the expiration date on your
Permanent Resident Card, or don’t apply for naturalization until your card has already
expired, you must renew your card.