Is the state space of a transient irreductible Markov chain infinite?

Yes. Recall that an irreducible chain is one where the entire state space $S$ is a single communicating class. In this case clearly $S$ is closed.

Further, recurrence/transience is a class property. So in an irreducible chain, either every state is recurrent or every state is transient.

It is a theorem that for a closed class $C$, then $C$ finite $\implies$ $C$ is recurrent (this is very closely related to the fact that you state).

So for an irreducible chain, applying the contrapositive of this theorem to the closed class $S$ gives: $S$ transient $\implies$ $S$ is infinite.