Sean was the one hung up on money, not Mary Kate!
This movie is very straightforward so I find it simply astonishing that apparently so many people have misinterpreted it. Firstly, this movie is set in a small rural village in Ireland in the year it was released (not the 1920s, for crying out loud). Basically, this means that the culture and customs of this village is Victorian to the core; think 1880s rural USA. This is not an exaggerated portrayal, but an accurate one. (Divorce in Ireland was illegal until 1996s; I love my Irish heritage, but Ireland by no means is a progressive country) This is why the character of Sean Thorton is having such a difficult time.
Because the culture he grew up in was so very different, he complete fails to understand what the actual issue is. Mary Kate's dowry (furniture and money) is the only thing in the world that belongs to her, made even more special that it was her "mother's and her mother's and her mother's before". She lives in her brother house at his pleasure. He has absolutely no respect for her and values her only as much as society dictates he must without losing face. Refusing to give his sister her money is all about his ego. Using the "conspiracy" as his excuse, he refuses to acknowledge his sister as a person of any value (and that he was only appearing to finally do right by her for his own ends i.e. marrying the widow). (Even her "lack of a fortune" before marriage shows how little he values her, he's the richest farmer in the county, but won't give her a dime as a decent brother would in this culture.)
Mary Kate tells Sean repeatedly that she needs her things because they're very important to her because of what they represent and because even if the money doesn't mean anything to Sean, it does to her brother. In a society where it is his job to defend his wife, he is allowing another man to disrespect her and him publicly. Despite discussing this with him multiple times, he is never able to really understand what she's saying because of his own Issues with money (which he repeatedly fails to explain to his wife).
It takes his wife leaving him for Sean to understand that whether he agrees or not this is not an issue he can avoid in his marriage or the society he lives in now. And it is only when Mary Kate runs ahead of him to open the furnace door that he truly understands it was never about the money.