The more desire we have the unhappier we are?

Well I suppose it depends on whether you can obtain whatever it is you 'desire'. 
In materialistic terms, a billionare who wants a plane, then a yacht, then a whatever, whatever, whatever and can afford all of it won't be unhappy the more he desires, because he has enough financial resources to obtain those things. 
I think the unhappiness comes from not being able to obtain the desired object/entity. Going back to our billionaire, a problem might come if he desires, say, a home-loving type of wife that doesn't require much and then can't 'obtain' that because exactly the opposite type of women seem to be attracted to him. 
Going now to 'ordinary' people who may not have the resources (mostly financial) to obtain what they want, yet are subjected to advertising of sorts (whether through movies, people they mix with, or plain adverts) that has the aim of making people feel they need something and tries to make them feel somewhat inadequate if they don't have it: 
in that situation, such a person, the more they desire perhaps the more unhappy they will be since they can't afford everything they desire. Wanting something you can't have generally makes people unhappy, since it's something you 'wanted' in the first place. 
I think desires regarding feelings work in a similar way. 
With every new desire a person might have comes a possibility that they can't have it, and therefore a source of unhappiness for the person. 
So...I would agree that (to put it this way): desiring less can potentially make one happier (or less unhappy). 
I think that the feeling of desire at least partially comes from feeling that what you already have materialistically or otherwise is inadequate/not being content with what there already is. 
Although there are various kinds of desires, some more destructive/less sustainable than others..it's a bit complicated. 
I'm talking about desire here as in wanting something, not desire associated with being in love with someone.