Apologies for the delay, indrcb! You should obtain a new form as it is very possible that the hotel will not accept the form if it expires on the checkin date. Hope this helps and have a wonderful trip!
If the expiration date is (example 8/31) & the stay is (example 8/31), the hotel should still accept it. It's similar (IMO) to when a bonus ends on (example 8/31) & the stay is (example 8/31), the bonus is still applied/valid as it's valid THROUGH the last day (same w/ grocery store coupons for that matter). If the OP checks in after midnight on the 31st (technically September 1st) it wouldn't be valid. If the OP checks in on the 31st before midnight it would be valid. However, to make certain it might behoove the OP to get a new cert.
And as long as we're into education mode, F&F certs are valid for 60 days (and can be used more than once during those 60 days - just print out the cert for the # of stays). But because of the 60-day expiration it's better to get the cert from the family or friend member closer to the start of the first 60 days - for example, if you book a F&F stay today for (example) December, don't order the cert today. Wait until late November or early December.