The Byzantine Empire's Italian territories were weak and vulnerable - it's difficult to see how somebody wouldn't take advantage of that.
They were made weaker due to a botched Gothic war, partially due to underfunding, and partially due to Justinians paranoia/Narses ambition - and then Persia to boot. A successful invasion of Italy would have to circumvent all of these - and as a result, Italia is much wealthier, and at worse is only a quick sail away in friendly waters (as the Vandals are gone and any other naval powers are in the Western Med, leaving shipping nice and clear). All in all, Italia becomes quite easy to integrate once conquered. Just don't burn the blighter to the ground first!
But let's say they hold on to Italy essentially in its totality and also have North Africa and all the other reconquests locked down for the time being. Latin is on its way out regardless - it doesn't matter how long they hold their western territoried, the Hellenization of the Empire is due to its core eastern territories and an ongoing cultural shift.
I'm not entirely convinced. With Justinians new law books in Latin, Italia back under control, and a Latin Emperor who will likely be able to leave a Latin legacy - I can see Latin lasting much longer, or maintaining its dual-status. Especially with a potential West/East religious split. I can imagine the language will split along those lines, with Latin being one of the two languages of state.
The Muslims are probably butterflied. I think the Sassanian wars go more or less as OTL, discounting butterflies. Maybe the Romans have more or less troops to use - either is possible. However, one thing that's important to note is that the migration of various peoples, including the Lombards but also the Slavs and Avars, is going to have a big impact on the Byzantines. I could even see the Avars managing to conquer large swathes of Italy, forcing the Byzantines back to the coast - either way the good old Eastern Romans will face increasingly difficult challenges throughout the early middle ages and will struggle to overcome them all - as in OTL.
If the Goths can be brought onside, then Italia is easier to defend, and could provide some reinforcement. But having the invasion resolve quickly means we can have Belisarius in the eastern theatre, perhaps even before the war takes place. Solid leadership, or at least someone who might predict an invasion could make the eastern war much less costly.
Adding territory isn't necessarily a boon to the Byzantines. It means more commitments, and I think you'll see that Iraly, ruined and depleted, is a net sink on the ambition of any Emperor - a vast province that needs Anatolian soldiers to defend and doesn't contribute all that much in taxes.
Again, if the invasion is quick and harmless as Belisarius had managed to achieve in his early invasion, then this isn't a problem.
If they lose Syria and Egypt and North Africa to some alt-Muslim expansion - or even just to Arab conquerors without unified religious affiliations - then Italy is probably on its way out too. What you're doing is creating a whole host of new naval threats for the Byzantines to react to. If an Iranian or Arab Empire gains a foothold on the Mediterranean, the Mare Nostrum becomes the Mare anyone who wants it. This will be bad for maintaining a coherent maritime empire - as Italy would be, given that overland connections are easily severed by Slavs.
Yeah, an alt-Arab invasion will be a nightmare for the Empire. They'll need to approach that problem carefully. But with Africa and Italia able to provide reinforcements, and hopefully a stronger more stable Roman Empire as a result - it would be easier to rebuff the Arabs. It all depends on whether the Roman Empire can appear intimidating enough to the Persians with their new territories and manpower to avert more wars, or win alternative wars. They could take a different approach and use the Arabs to invade Persia - allying with them to set up an Arab kingdom in Mesopotamia. After that, the Arabs form a buffer between Rome and Persia.
Edit: Italy is only strategic depth insomuch as you could flee there if Constantinople is lost. But if Constantinople is lost, the resulting Empire is going to be very different. Constantinople anchors the Empire - holding it pretty much together no matter how much of the interior is lost. Can't get much better than that. When the Bulgars overrun the Balkans or the Turks overrun Anatolia, you have the perfect base to stand and defend... until both are overrun and you don't have any manpower.
*applause*
Italy isn't just strategic depth. Much as North Africa provided the Caliphate a new demographic and economic base to invade from - Italy can do the same for the Romans. If it isn't going ont he offensive, it then provides a second base for invasion. Which means any northern invasion into the Balkans is at risk of a Roman army from Italia cutting off their logistics, and invading their home territories. (I'm looking at Pannonia/Hungary/Avars here).