I had to chew on this one for a little bit. When any of our "things" can be replaced with the press of a button, we'd probably adopt a cultural of (voluntarily) loose ownership. Where what we created for ourselves we would still say we own, but most of us would be willing to give away most kinds of things if someone asked. I wouldn't call that communism, I would call that one of the effects of a rising tide lifting all boats: an increasing generosity.
Now if you're talking about an abusive (i.e. controlling) state, where the people are deprived of all the replicator technology and it's only in the hands of the government for it to dole out the output of, then also assuming a near-infinite energy source to run and I guess be the raw material for these machines, you could say that that thorny issue with Leftist economics of "eventually running out of other people's money" would be overcome.
I guess in that sense it would "work".
p.s. An interesting (to me at least) thought just occurred: Mr. Roddenberry didn't swap Right-wing economics for Left-wing economics in his made-up universe, he did away with economics?