First of all, define "didn't happen". Is it delayed a couple decades due to slightly different evolution/failure to transfer from rats to humans immediately, in which case the problem is being kicked down the road? Does Yersinea Pestis simply not come to be, for whatever reason (because that is an evolutionary PoD, which, although I think it should go here, belongs in ASB)? Or does it burn through its early hosts quickly, saving countless lives?
Regardless, the effects are far-reaching and huge. I tend to agree with the gist of Pierson's claims, that the Bubonic Plague was very much responsible for the Fall of Rome. Without it, the Gothic Wars will reach a conclusion sooner, and recover quickly. The Balkans will not be too depopulated, meaning Slavic and Germanic invasions are repelled fairly easily. The Persians are stronger too, of course, but as we saw IOTL, the Romans were mostly effective at holding them off, barring disaster (which was brought on in no small part due to the Plague).
However, things will not be easy. Rome is bound to get very cocky, and even if all their frontiers are secure, push for more. Spain is the logical next target, knowing how ambitious Justinian and his successors were. So the Roman resources are still going to be strained, as they defend in Syria and Mesopotamia, pay off would-be invaders and allies, defend the Danube and Italy, keep the Berbers under control, and finally now another expense in fighting wars in Spain. Maybe it will work, and Spain will be fully conquered like Italy or North Africa, but in that case it's possible they go for even more.
I don't think it's inevitable that they overextend and ruin themselves, though. It cannot be overstated how many people, and just as importantly, taxpayers, were lost due to the Bubonic Plague. Rome was the hardest hit, but they were not the only ones- Persia got it badly, too, as did what Urban areas existed outside the borders of Rome. And outbreaks continued for ages, stunting population growth and contributing to de-urbanization and instability. It's an easy guess to say that life in general in Europe and the Mediterranean would be a lot better without the Bubonic Plague, with more people, more wealth, and more minds creating new technologies and advancements.
So I guess the question becomes: Is Pax Romanica inevitable? And the answer is of course no, but it is likely in some form. Even if that form is a Rome inhabiting the borders of the Eastern Roman Empire+Italy+North Africa. The Arabs will be unable to do anything against the fully populated Sassanid and Roman Empires, that is, if butterflies don't stop them first. And instead of being perpetually in debt, the Emperors of Rome will have a large tax base to exploit and keep things together. This doesn't mean they'll restore the Old Order, or that it will last forever, but it's certainly going to be a lot more likely than OTL.
It's hard to say much else without knowing what the butterflies will do, but it is a fascinating PoD.