If you have a big budget, there are beautiful resorts, spas, carriage rides and other typical romantic offerings available. Disney also has a florist so you can fill your room with roses, if you'd like. Info on Carriage rides:
http://www.buildabettermousetrip.com/activity-horses.html
Restaurants - There are some great dinner options like going to the California Grill (Contemporary resort) for dinner during the Magic Kingdom fireworks. This restaurant has its own viewing platform w/ the Wishes music piped. Other romantic options include "Victoria & Alberts" and Narcoosee (try to get a table by the water). These are high end options. Dinner at Cinderella's Royal Table (Magic Kingdom), San Angel Inn (Epcot), Brown Derby (Disney's Hollywood Studios - although my husband and I actually prefer Mama Melrose) and Yak & Yeti (Animal Kingdom) are also good choices.
The "normal" romantic things make me feel pressured and stressed so booking a night in a lovely room at the Grand Floridian and then doing a nice dinner would be completely counter-productive. My idea of romance at Disney World: spending the day at Epcot talking, holding hands, and enjoying all the live entertainment in the World Showcase. Strangely, one of the "most romantic" Disney meals I've enjoyed was when we got counter-service food from the Yakatori House in the Japan pavilion and we ate it in the outdoor Japanese garden after dark.
One time, the ride broke down on Spaceship Earth (the silver ball at Epcot) and we were just stuck there, alone in our ride vehicle for about 30 minutes. We engaged in the kind of making out generally reserved for the back of school buses but you won't want to go any farther than that because Disney has cameras everywhere. But ride break downs are hard to pre-arrange.
A few more things my husband and I find romantic:
1.) Floating around the Lazy Rivers together in the water parks. We also find the double tube slides and watercoasters romantically thrilling.
2) Renting a Kayak (you will get a little wet) at Port Orleans Riverside resort or a canoe at the Fort Wilderness campground. The leisurely pace and the quiet paddling lend themselves to good conversation. The Sea Raycers (motorized) can be kind of fun to - especially in the Epcot resort area and in Downtown Disney. The Sea Raycers have speed governors on them so don't think you're going anywhere fast.
3) Seeing Cirque Du Soleil - La Nouba in Downtown Disney is a fun splurge, especially when coupled with dinner.
4.) Strolling through the resorts - especially the Animal Kingdom, Boardwalk, or Polynesian resorts. These three are all great after dark.
5) Picking out a couple of desserts from either the France, Germany, or Norway pavilion counter-service restaurants (Epcot) and enjoying them together while we wait for IllumiNations. (Be sure to get forks & napkins).
6) Staying at the Magic Kingdom until it closes. The shops stay open for about an additional hour and it is wonderful to enjoy stroll along window shopping. looking at the lighting on the castle etc. There's no advantage to hurrying out - you'll just wait in a line for your transportation - might as well enjoy the park.
7) Watching the Electric Water Pageant - a nighttime floating parade - from beach area at one of the Magic Kingdom resorts. We especially like the Polynesian resort - then we catch the water launch over to the Magic Kingdom, just for the ride. Electric Water Pageant info:
http://www.buildabettermousetrip.com/activity-electricwaterpageant.html
8.) Another low-cost gem: Coronado Springs Resorts has The Pepper Market - one of the nice places to use your
Disney Dining Plan counter-service credits . Sleep in, then find your way over to the Pepper Market for a nice lunch. After lunch (this is one of our favorite things!), go out find one of their 2-person hammocks to lay in together. You'll probably fall asleep, especially if you've been hitting the theme parks really hard so be careful that you don't get sunburned to a crisp.