We honeymoon'd at the GF in 2001. I'd like to second the idea of the Boardwalk, perhaps the flying fish (for a signature meal) or spoodles for a slightly cheaper meal (its a great non-signature dinner though). Then a hand in hand waterfront stroll and an ice cream at the board walk or at beaches and cream at the beach club. If you like the idea of an old fashioned east coast board walk night, this can be very romantic. They have two sets of entertainment that comes out to the boardwalk at different time intervals. Very charming. Wife and I love this! You can see the illuminations fireworks from the boat pier entrance at the Yacht club, or even a few other places around crescent lake.
If you want to stay at the GF, can I also second scrapping pennies together for Victoria and Alberts. You will NEVER forget it. Let me promise you that, I've had the luck and pleasure of visiting each signature disney restaurant at least twice and we are glad we went for V&A on our honeymoon. You can keep the cost down by not opting for the wine pairings. The lady will get a rose, I think we got a nice chocolate plaque to commemorate our honeymoon.
That being said, we really like Citricos (which can get pricey also), and two of our most memorable meals over the last 6 years have been there. It is a bit more quiet and sophisticated than some of the other restaurants. We had some nicely hand made pastries given to us in honor of our anniversary last year.
Also another ditto, Bistro de Paris, they have a new chef as of Septemberish, and he seems to be off to a good start. We got a late ADR, and our server eventually moved us to a perfect seat for illuminations. It's very quiet and romantic, and the service is very good at what they do. Just off the chart marks here. The fixed menu was an excellent choice.
One of the things we usually do is to dress up a little for our "big" nights out. We plan a bit more time to get cleaned up before hand and go "out on the town". Makes us feel a bit more special and magical. For V&A you will need to as they have a specific dress code.
As a night out, on our honey moon, we hit an early diner upstairs at Wolf Gang Pucks Grand Cafe, and caught the Cirque show. It's a nice equivalent of a night in New York. After the show you can stroll downtown disney until it slows down. Although costly, the show, I think is excellent, and I was glad we decided to go for it.
Speaking of slowing down, some of our most romantic moments have been after the parks close and clear out. For example when you eat at Bistro by the time you are finished the park is closed, and you exit the building totally alone. It's weird to have the park to yourself and be able to hold hands and stop and look out over the glistening lights over the lagoon.
Late night use of the quiet pool and spas at the hotel.
An illuminations or Wishes cruise (I know expensive...haven't swung that one myself). Maybe share one with someone on the internet.
Check out the Perlmutters book (Disney World for Couples).
Walking the beach at the Poly during Wishes or during the Electrical Water Pagent. Grab a beach chair lounge cover chair thingie.
If you do eat at the Cali Grill Don't forget to go out on the cat walks to look out over the MK, hopefully you'll be alone in doing so.
A nice meal at Artist Pointe with an after dinner walk around the Wilderness Lodge, it is one of the best themed properties, and at night can transport you away
the running water, and the lake front.
Yachtsman Steak House has been quieter and intimate for us as of late (notice the premium we put on quiet), if your in that area for a fancier meal.
The Grand Floridian did have a custom menu and with enough ahead notice they could have a meal for you at a table just about anywhere on the grounds. It's called the "Private Dining Menu". Expensive like room service, but if you want to be alone....I'd rather go to V&A, Citricos, or Narcoosees myself.
A lot of the more secluded areas are romantic, the garden area in England is nice to relax in, especially if you like the Beatles music. The Japan garden is beautiful. Exploring the EPCOT countries, every nook and every cranny just feels more adult and more romantic in nature.
If you happen to be coming home to the GF later at night on the Monorail, ESPECIALLY if you are alone, ask to be seated with the Pilot up front, its really cool.
Never under estimate the power of the late night ice cream! Or Chocolate anything
Wherever you eat, if you can eat later, it will make it quieter, and you will have a better chance of being pampered. Also, after, you can explore the property. We like staying in the parks late, take a last ADR before closing, and close the place, walk out almost alone. Watch the fireworks from more secluded places, beaches, etc.
I wish you the most magical time! I wish I could relive mine,
George
from another Disney 
! May I suggest:
I would really like to try Citrico's. I heard a lot of good things about that restaurant. Of course I plan to request the same room I proposed to her at the GF, though it was only a pier view, it has special meaning now. 
