Can I give a gift card an expiry date?

Yes. Go to Plugins → Gift Cards. In Issue a gift card, set Expires on (Optional) to the last day the card can be used. Fill in the Amount and anything else you need, then click Issue gift card.

Leave Expires on empty for a card that never expires.

If a customer tries to redeem the card after that date, they see "This gift card has expired." and no credit is added. Customers should therefore turn the code into store credit (My Wallet → Redeem a gift card) before the expiry date.