Does an irreversible thermodynamic process always increase the entropy? I know that if over the a given thermodynamic process the entropy of the universe increases, then that process is irreversible. But does the opposite also hold true, i.e. do all irreversible processes increase the entropy of the universe?
 A: 
But does the opposite also hold true, i.e. do all irreversible
processes increase the entropy of the universe?

Yes, all irreversible processes increase the entropy of the universe where the entropy of the universe is defined as the entropy of the system plus the entropy of the surroundings. Moreover, since all real processes are irreversible, because reversible processes would take an infinite amount of time to complete, the entropy of the universe is constantly increasing.
Hope this helps.
A: Gibbs explicitly defined entropy as the quantity that always increases in irreversible processes. This was the way to formulate a coherent set of thermodynamic laws. In this sense, the thermodynamic entropy increases in irreversible processes by definition.
On the other hand, for Boltzmann entropy and its varieties used in statistical physics, this property needed to be proven.
Note also that, while the net entropy of the universe (the system plus surroundings) always increases, the entropy of the system itself may decrease (unless it is isolated) - living organisms are a good example.
