well, this is one of those difficult questions to answers. i myself have been trying to figure this out for few years now.

some places that i can suggest are:

new cafe at the corniche near the Asian games mascot - its a lovely place to be in the evening with your luved one. now the weather is nice as well. take a table facing the sea. but weekends can be too crowded

check out the restaurants at the renovated souk waqif in doha city. some restaurants are quite nice with a romantic setting

beach at the sealine resort followed by dinner. again might be difficult to get a private space during weekends.

corniche during late evening (family area section) with bbq. try to get a lawn area with not much nosy people around.

probably a movie at the citycentre cinema?

and some guys mentioned above, maybe a long drive to Ras Laffan or Dukhan with a speed limit of 40 :-)

sheraton beach side restaurant - quite a good ambience

hope this helps.

have a luvly romantic weekend.
kumar