The issue of utopianism in Star Trek is not a straightforward as people tend to discuss. It is a key aspect of the franchise and has been worried over in a variety of forms.
In TOS, Paradise was seen as a dangerous idea which saps the individual's desire to achieve and/ or creates twisted societies. (see This Side of Paradise, A Taste of Armageddon, The Apple, The Return of the Archons, etc). At the same time, it promoted an obliquely referred to ideal of humanity no longer motivated by greed and selfishness. This was extremely of its time - in the 60s the hippie ideal was fighting against American can-do-ism. Star Trek managed to encompass both and tried to imagine how they would work in concert.
By the time of TNG, Roddenberry had fended off numerous questions from fans over the years about how this "New Humanity" worked. He shifted over to a much more utopian vision and you end up with a money-less society (in the 80s this was the clearest counter-culture dream you could present) that functions based on the assumption of cheap, clean and unlimited energy (in the 80s this was the clearest straight middle-of-the culture you could present).
Now consider, if cheap, clean, unlimited energy were actually available, it would deeply change the dynamic of the market economy and would have far-reaching consequences on human society. So, it seems to me, this is the real question at hand. Not - would people actually work for no money? In general I think a society of cheap, clean, unlimited energy would have a lot of people who would work for idealistic goals. We already see that, even in a market economy of (relatively) cheap, dirty, limited energy. But there would be a large group of people who would mostly just enjoy themselves with shallow past times as well. We also already see that. Star Trek has no interest in these people, and only shows us, honestly, the nerds of society. And I use nerd in its highest and most complimentary form (I personally never saw it as an insult - oh, I like to learn and do/ create interesting stuff? Yes, I do and thank you.)
DS9 was decidedly critical of the utopian society presented in TNG - most especially through the outsider voices of Quark, representing the proud capitalists among us, and Garak, representing the devious shadow self of Star Trek. Again, this captured the counter-culture/ conformist split in 90s culture. You see the first truly diverse set of characters, not dominated by white humans, but that diversity creates uneasiness, and eventually extreme conflict.
The other shows are weak in many ways because they lost track of how to embody this key aspect of Star Trek - capturing the times and struggling with the idea of what humanity would do if utopia was in its grasp. It's not an easy question. TNG, which seems to be dominating this discussion, was only one of Star Trek's answer to the question.
Whether a society has money or not is just a side effect of utopia. Energy is the key and human society's evolution has revolved around civilizations and then individuals being able to leverage more and more energy. Energy must be expended first to meet survival needs. Once that is accomplished (and Star Trek does put forth that this has been achieved), philosophers are very likely to tell you that humanity would then set about trying to meet
Maslow's hierarchy of needs. So, Star Trek is probably right that once everyone was fed, clothed and sheltered, people would start looking for things to achieve. Money has little to do with it, since money is only a means to these much larger ends.
