The question and the answer by user14471 seem to relate to the issue of ergodicity in economic systems and models. Economic systems (in reality) cannot conceivably be ergodic, while some economic models are ergodic (those that do not attempt to reflect any of the non-ergodic properties).
Necessary concepts to answer this question: Ergodicity, microstates, macrostates
Ergodicity is the property of a system to spend approximately equal amounts of time in each of its microstate (if you observe it over sufficiently long times). A microstate is simply the state of a system if you consider every property of every element of the system. Not all microstates are distinguishable. Undistinguishable microstates constitute a macrostate. That is, all macrostates are distinguishable but some have more microstates than others. An ergodic system will be more likely to assume those macrostates with more associated microstates (that is, the macrostates with higher entropy).
Consider a non-economic example: gas in a container. For the definition of a microstate, the position of every molecule is important, for a macrostate only the distribution of the gas matters. While there is a macrostate in which all the molecules are crammed together on one side of the container, this one is extremely unlikely; a macrostate with an even distribution of molecules across the container is much more likely. Since the gas can, however theoretically assume each of the microstates in the future, independent of which other microstates have been assumed in the past, the system is ergodic.
Now consider an economic example: the distribution of dollars across the population. A microstate considers which dollar is owned by which person, a macrostate only considers the distribution. While microstates are not distinguishable or even measurable (how do you distinguish the different dollars in your bank account?), the number of microstates per macrostate is important. Theoretically, all dollars could be owned by one person (while everyone else has nothing). This is an extremely unlikely macrostate, while a more even distribution across the population is much more likely. Actually the likelihood (and entropy) will be maximized by a Gaussian distribution. (However, if you measure the distribution of wealth, you will find that it is not Gaussian but heavy-tailed.)
How ergodicity, irreversibility, and the arrow of time, are related
Ergodicity can be phrased differently as: every microstate is reachable from every other microstate. It is reversible, it does not have an implicit arrow of time. It is easy to see that this is not the case if the system has an attractor (a stable equilibrium) that captures trajectories (development paths) which will then not be able to leave the attractor again.
Ergodic and non-ergodic models in economics
Armed with these concepts, we can now return to the question of reversibility in economics.
Models of simple exchange of goods etc. are ergodic. Depending on how and according to what prices and preferences the goods are exchanged, every microstate of the system is reachable and possible. What is more, every transaction can be reversed.
In the case of models of growth, developent, or technological change, ergodicity will typically not hold any more (unless you can have degrowth such that the system reverses its growth trajectory exactly). These systems typically have attractors (Solow-Swan models for instance have an attractor once you remove the neutral technical change term) they have production functions that would generally not be considered reversible, and they may not allow for recessions and reversion of technological change. More complex models of technological change and development from evolutionary economics or so will definitely
That said, economic models have a certain tendency to assume ergodicity for whatever is not part of the model. Macro-models (including Solow-Swan models) assume that the structure of the microlevel (that is not modeled) will not interfere with how the model works, that agents are exchangeable (representative agents) and that transactions are neutral. For RBC and DSGE models this is made more explicit by assuming an unbiased i.i.d. stream of shocks acting on a largely homogeneous population of agents. Some varieties (HANK etc.) try to address this, but the heterogeneity those models allow for is extremely limited. Agent-based models address this from a different angle and are able to achieve the non-ergodicity property observed in real economic systems (see last paragraph below), but they have their own problems, as was discussed in a different question.
Ergodicity in real economic systems
In real economic systems, it is obvious that development paths are not reversible. You may suffer a recession, you may even experience an extensive collapse and lose advanced technologies, but you will not be able to dismantle and sell off the associated capital goods (and the human capital) in a way mirroring how they were acquired. Further, humans react vastly different to gaining and losing wealth (loss aversion).
And beyond this, the very distribution of wealth - a heavy tailed, Pareto distribution, not a Gaussian - will show you that the system does not achieve (or even come close to) the entropy maximizing macrostate with the most associated microstates. This is a structural property typically found in complex systems with self-organizing properties (the issues Prigogine wrote about as mentioned in the OP). You will find similar distribution also among firm sizes, regional agglomerations (city sizes), and a ton of other things that are more or less related to economic systems.