One of the reasons the Balkans are full of Slavs is that the Plague (and consequences) left them more or less empty. Without the Justinian plague (or even if it happened at a slightly different time), you might have a MUCH larger Illyrian-speaking part of the Empire there, possibly Roman Catholic.
I'm not quite sure about that- I think that the communities of the Balkans, while certainly damaged by the Plague, did not simply get wiped out and replaced by Slavs- there were, for example, Latinate speaking communities in the Danube region in the reign of Basil II. I think it's more likely that local people adopted Slavic dialects in the seventh and eighth centuries to cope with an influx of a new governing class that perhaps numbered perhaps 30,000-40,000. On top of that, there are the Turkic speaking Bulgars, of course, but they're another issue.
Regardless though, the plague changes this- I agree that the Balkans will not become Slavicised. How many people were still Illyrian speaking by this point- I'd venture to guess that by the sixth century, Latin was the primary lingua franca in the Balkans away from the Aegean region.
How close was Europe at that time to reaching its maximum sustainable population given the agricultural capabilities of the day? Crop rotation and better plows don't come along until the second millennium, so how big could Europe's population have gotten without them?
I feel like that'll have a lot of influence in terms of determining the more long-term effects of removing the plague.
I know Syria and Egypt were at a peak of economic growth pre-plague, not too sure about Europe. I'm guessing that the confusion following the collapse of the WRE, and the beginnings of the slow breakdown of centralised power in the successor states saw land perhaps becoming less productive than it had been, due to the decline of the powerful Senatorial landowning classes. I'd guess that Western Europe in 540 was operating below the level of output that it had reached in the fourth century, but was nevertheless doing much better than it would be doing in, say, the eighth.
Regarding a "no plague" scenario, the consequences will be vast. Without the plague, I'd expect to see the Gothic wars ended more quickly in favour of the Romans, perhaps by the later 540s, which means Italy is rather less devastated. The post-plague Roman army is quoted at having a manpower total of 150,000 set against a fifth century level of 645,000. Even if we discount the armies of Gaul and Spain from this equation, since these are areas I'm guessing Justinian will not attempt to conquer, it still gives his successors an army of perhaps 300,000 to play with, which more or less kills off the Slavic and Lombard incursions. A major Persian breakout is much, much less likely, although political problems don't rule it out.
A strong reassertion of Imperial rule in Italy kills off the idea of the Patriarch of Rome becoming anything other than a puppet of Constantinople- uppity Popes will be removed and mutilated. I'd therefore expect to see national Churches developing in Francia, Britain and Spain, and perhaps these regions developing their own branches of Christianity. Speaking of religion, the fact that the Pope is now a vassal of the Emperor makes the search for a doctrinal compromise in the Roman Empire as a whole much, much easier.
The Empire itself will largely remain substantially as it was in the 300-600 period. Greek is unlikely to become the language of the state, and many quintessentially "Byzantine" things will never emerge- this will be an unarguable Roman state through and through. It will continue to raise taxes to fund an army based on heavily armoured troops, and to base its administration of a network of large cities which will have never shrunk. Doctrinal and political difficulties will continue, of course, but unless the Empire encounters some serious bad luck, the system will continue to run reasonably well. There will be the occasional disaster on the Persian front, but in general, the situation will remain calm.
The Mediterranean trade lanes remain open. Islam is butterflied entirely. The Romano-Germanic successor states may be able to continue to operate as miniature Roman Empires, levying taxes and raising standing armies- this obviously means no feudalism. Development of Europe beyond the Rhine and Danube will obviously turn out in a completely different fashion- I can see Slavic and Turkic kingdoms emerging in what is now Hungary, Romania and Slovakia, which will gain their influence from the still prosperous Roman world to the South.