Strangely, it does. It's not all, but it's a more or less 30% variations of your mood, given how it impact your life. Tho', being reasonably wealthy, and filthy rich, have no difference. There is a ceilling at how money can improve your happiness, which is not that high. To be at 10k/month or 1M/month dosen't matter after that, for you mostly have the financial security and flow to do mostly what you want, on a casual level.
So... In brief, yes, the amount of money you own determining your way of life, and quality of life, is a part as a factor of how happy you are. The twitter dude isn't wrong on all of what he says, a hard worker in a bad financial situation will be less happy with adultery than Will Smith. Does it mean Will Smith can brush it because he is rich ? No... But he won't have to add that to other problems a working man could have.
Again, that being said, the twitter dude is a douche, thinking it's the only and sole factor and that "being rich means you can't be sad".