When carrying out a molecular dynamics simulation, one has to choose an ensemble. Depending on the problem, one usually works in the microcanonical (NVE), canonical (NVT), or grand canonical ensemble (NPT).
In the NVE ensemble, maintaining reversibility of the simulation is very easy, as the total energy is conserved, so simply starting at the endpoint of the simulation with reversed momenta should reproduce the initial trajectory but in reverse.
In the NVT and NPT ensembles, one has to maintain a constant temperature and/or pressure. Taking NVT as an example, there are many different ways to keep a constant temperature using different thermostats. Thermostats are all methods of keeping temperature constant by exchanging heat with a hypothetical heat bath. There is a range of complexities of thermostats and there's no need to discuss the subtleties of these here. The important point is only that some commonly-used thermostats, such as Nose-Hoover chains, are deterministic and hence result in reversible dynamics. On the other hand, there are Langevin thermostats which are very effective and efficient but are stochastic. So, short of playing a pseudo random generator backwards, the dynamics are not reversible. (I've never seen that reversing a random number generator thing done, but I think it's at least possible in principle?)
I've seen it mentioned many times in the literature that Langevin thermostats result in dynamics which are not reversible. Because I've seen this mentioned so many times, I've always believed it to be important, but I have no idea why one would care about being able to use the end of a simulation as new initial conditions and then propagating the simulation in reverse.
So, why does the reversibility of a molecular dynamics simulation matter? In what situations does using a deterministic thermostat/barostat provide a distinct advantage over a stochastic one?