I depends on the program so you'll want to research each program individually. This information will be in the program's terms and conditions mailed with the card. But generally they break down into two categories:

Cards that earn for another company's program

These include Citi American Advtange, Chase United, etc. Since the rewards earned on these cards are transferred each month to the frequent flyer program closing the credit card sponsoring account generally does not impact the awards as they've already been transferred away to the airline or hotel program. Many will advise waiting 6 months, esp w/Chase, before closing but reports of card companies taking back awards from airline/hotel programs are not common.

Cards that earn for their own reward program

These include Chase Sapphire, Capital One Venture, Citi ThankYou, etc. Since the rewards are with the company program of the credit card company, closure of the sponsor credit card usually means loss of the points. Some will have a X number of days to use. Most will delete your points when you close. General rule is to use them or transfer them before closing.

These cards are a good reason to never close an account in anger as well. If a membership fee is creeping up on you look for a no-fee version of the card you have with the same rewards program in order to maintain the balance.