I would not say so much any rides that come to mind as much as locations around WDW. The rides are too short, public, and loud to be romantic. You get semi-privacy in the HM, for example, but I'd hardly call it 'romantic' especially with the 'attic' scene where the disappearing husbands. Unless creepy is your idea of romantic (as horror movies are considered 'romantic' by some people.)
SEcond, I'll preface everything by saying there areally are no 'private' areas in WDW- except your hotel room interior. Some resort balconies are closer to private than others, but many are not really private at all.
There are rose gardens in Epcot and MK, the ones in MK are more secluded.
The area around the Boardwalk, esp at night or when passing a wedding are nice. The beaches in the eveng will likely be empty. Public, but empty.
Dinner at many signature dining locations can be romantic. See above and pair FLying Fish with a walk.
A balcony with a fireworks view is romantic during the fireworks, esp. Likewise the balconies at AKL would be romantic - even the ones that are not private. Rarely do folks go out there, even during the day. Though folks inside may be able to see you. At night the resort has a campfire story hour. Again, not alone - but romantic. (again, to reiterate my point above - all of the CR fireworks view balconies are public from a vairiety of places. Likewise with all Swan/Dolphin balconies. I'm guessing just about every balcony has security cameras of some sort.)
After dining at Cali Grill you can go out on the balcony. (ask) You won't be alone, but it's romantic.
POR and FT.W/WL have carriage rides that would be romantic.
The peoplemover in MK is - er - notoriously 'romantic' -so to speak. I don't find it especially romantic, but you do sort of get a sime private space. You are not alone though and for much of theride you are being watched. This is true on most attractions. But even there, the attraction makes some fairly sharp turns at several points.
Any fancy dinner at GF. Just about any place there except the character meal is romantic. The quick serve is not especially romantic per say, but you can take your food outside. Depending on how crowded it is, you get a nice view of castle from there. You can also walk along the whole 'shore' of the resort. though again you will be able to be seen.
Another idea is to rent a private fireworks boat.
Another idea that may or may not work is to take one of the many shuttle boats around WDW. The bigger ones probably are not likely to be all that romantic per say, but you could very likely end up on one of the smaller ones an have the boat almost to your self. I'm thinking most specifically about the ones that run between CR/Ft.W/WL.It's also possible that you'd et aboat mostly to yourself between MK and GF/Poly - but the first is more likely. the boats in the BWK are are bigger and usually more cowded. There's also smaller boats between SSR/DTD or POR/POFQ and DTD. I haven't been on thoe enough to comment much, but maybe you'd get one to yourself.
Of all resorts, you may also want to conside POR (mansion side only!) or POFQ. Then again, walking CSR at night is pretty and quiet and romantic. They also have a nice bar - Cafe Rix for adult beverages.
I was thinking of the pitch black section before the "drop" you could steal a smooch or two 
