Ellen,
Yes, they could refuse to accept it . At a minimum, it becomes a hassle and extra work for them /delay for you to have the ticket corrected.
I have redeemed many ( higher end) restaurant coupons and Groupons, and I've never had a problem with lesser service. The whole point of the coupon is to introduce you to their business, so it would be very short-sighted of them to treat a coupon customer as "less". The kitchen won't know about a coupon, so there is no risk of a small portion. Just make sure to tip on the full amount that you would have paid without the coupon.
Deb