I have heard that Disney is starting to crack down on unauthorized use of discount codes, though I have had no personal experience with this.
This may differ if you book through AAA. I would not take the chance of having my room changed to rack rate. Bring the card. I've booked AAA rates in the past but don't remember if I was asked or not.
If there is a problem with your reservation when you get there, you will really want your card.
Doubt if you'll need it if you booked thru an AAA office - they would be the ones verifying your membership in order to book the trip in the first place. Our package was thru AAA and no one at WDW asked for our card (altho we had it as it's something I always have - never know when you'll need emergency road service )