Drama or movie. Japanese/korean. Ive seen a millionaires first love. Totally loved it!
I guess I have seen only one of this sort: Tatta Hitotsu no koi - here:
http://mydramalist.com/japanese-drama/56/tatta-hitotsu-no-koi
Very rich girl, poor boy, social differences, family interventions - it was quite nice.
I haven't seen much of those kind of plots. Only similar drama that pops in my mind is Material Queen. Unfortunately it isn't Korean or Japanese but Taiwanese. Girl sacrifices rich lifestyle for the one she loves though not right away.
Secret Garden (korean) - later on in the show the lead male decided to give up his wealth for the girl he loved! Though in the end he wasn't poor, but he wasn't as rich...fantastic drama! Birth of The Rich (korean) - this came to my mind when u said 'money'. I can't really remember how it ended...but I know the girl came from a wealthy family and rebelled against her father to be with the guy. Pretty good too, the first half is a little slow, but the second half really hooks you! :)
The Princess' Man , probably not exactly what you r looking for, but it fits your criteria
Me Too Flower - Guy gives up his wealthy company in the end and plans to live with the girl instead.
City Hall :) Although it takes a while to notice what the male lead is resigning to, and Sungkyunkwan Scandal
New Tales of Gisaeng (Da Mo to Sa Ran very touching scenes, if you like touching scene in Princess' Man this drama have comparable scene)

Stars Falling from the Sky
Taiwanese/chinese "waking love up" !!! "king flower" aka "substitute princess" "fated to love you" "miss rose" "smiling pasta" japanese "rich man, poor woman "
Spoilers ------- fondant garden (warning Korean actor dubbed) the male lead pretends to have lost his memory to live a poor lifestyle to be around the girl he likes and to live normally for half of the series Dal ja springs - at the start of the series lee min ki had already runaway from home and quit being a lawyer in order to live his dreams but ends up going back so the woman he loves doesn't struggle only to give it up again
Flower boy ramyun shop (korean)... though it happen at the last part..
Birth Secret has a situation similar to this at the very begging of it. But it also has amnesia too so you might not like that aspect. It's a rich woman poor man situation.