Irreversibility is more a practical than a theoretical concept, in my opinion. 'In theory' all reactions are reversible. Take this highly schematised reaction:
$$\ce{A + B <=> C + D}$$
We define the equilibrium constant as:$$K=\frac{[C][D]}{[A][B]}$$
[Using Nernst][1] we can now establish a relation between $K$ and the left-to-right change in Gibbs Free Energy, $\Delta G^0$:
$$\Delta G^0=-RT\ln K$$
(Of course, as you know $\Delta G^0=\Delta H-T\Delta S$)
Evaluate this Nernst function to conclude that the more negative $\Delta G^0$ is, the higher the value of $K$:
$$K=\mathrm e^\frac{-\Delta G^0}{RT}$$
For very negative values of $\Delta G^0$, $K\to+\infty$ and by the equilibrium constant equation, the concentration of the reagents at equilibrium is essentially $0$. Then we can write:
$$\ce{A + B -> C + D}$$
Such a reaction we would call irreversible. There is however no clear cut-off point and many reactions will have $K$ values that somewhat defy categorisation with respect to being 'reversible' or 'irreversible'. [1]: https://www.chem.purdue.edu/gchelp/howtosolveit/Thermodynamics/K_from_DelG.html
