Can I redeem the value an out-of-date gift card?

No you cannot because it is already expired, and the place will not take it because of it having expired. Unless they're good about it, they'll give you an excuse like "it's invalid or "it's expired." However, there is a movement in various consumer protection groups to get rid of expiry dates on voucher cards in several countries, so things are changing. It can't help to ask to see the manager, to explain the situation and to find out if the manager/company is prepared to honor the money already paid to them. Mention that consumers are wanting this sort of service, not an expiry date.