It has gone quite far, in fact today Finley has been pretty thoroughly superseded. His importance in the field is unquestioned, but most of his theories and perspective been heavily revised if not entirely disregarded. Perhaps the largest single factor is that whereas Finley tended to disregard archaeology's contribution to the study, today it very much takes center stage.

Probably the major theoretical change is that Finley's reliance on Polanyi's theory of substantivist economics has been in many ways either discarded or chipped away, and today most focused studies implicitly accept a market model for economic analysis. So, on the whole, the field has been moving away from the Mauss-esque anthropological models. However, a recent and very powerful critique of the market model surfaced in Peter Bang's The Roman Bazaar, which essentially posits that a market model is unsuitable for the sort of large "tributary empires" as Rome and (in his comparison) Mughal India. The book contains a number of empirical shortcomings, but as a whole it provides a really interesting path that is probably only going to get more prominent. It also made very clear how much a more social science approach is needed, particularly in regards to neo-institutionalist economics. I don't mean to give too much prominence to this one book (other recent ones I could mention would be The Ancient Middle Classes and Trading Communities in the Roman World), but it nicely exemplifies the sort of point the field is at now, where the old primitivist/modernist debates Finley was neck deep in are being discarded as inapt and uninteresting.

If you want this is a more coherent, detailed and learned form, Walter Scheidel has a very nice summary article. For a best text to read, what are you most interested in? My standard response is to go for Archaeology and the Roman Economy because it was written specifically for an interdisciplinary audience.