Joe Smith rolls his money from an IRA into the ABC Retirement Plan. ABC Retirement Plan only allows for hardship distributions prior to separation of service. Is Joe's rollover monies subject to this or can he take it at any time?

You said the plan does not allow the distributions. That is your answer. The plan can be designed differently.

Read carefully. Many (most?) plans that allow rollover CONTRIBUTIONS permit distributions at any time because these distributions are not restricted under law. To us, allowing rollover contributions but not unrestricted distributions is inconsistent.

Following up, if a plan specifically does not allow any inservice distributions, as per the document, then is it safe to say that includes the rollover contribution that was moved into the plan?

If the plan doesn't specifically allow in-service distribution of roll-in contributions, no in-service distribution is permitted.

So, yes, "it safe to say that [no in-service distributions] includes the rollover contribution that was moved into the plan."