I disagree, especially in the long term - there isn't anything to gain unless there is a complete and total victory over the Caliphate, the Romans and Persians couldn't do that to each other and still hold the conquered territories. They're both too exposed. Especially as it is likely that the Romans would lose the Ghassanids as clients. (In fact, I think they already did).
The obvious targets are in the Caucuses, Syria and Mesopotamia. Persian Syria and Roman Mesopotamia would both be far too difficult to hold, especially with a third foreign power on the border, whilst a Caucasian conquest just doesn't have the same value. Regardless of who won, the Arabs could just swing in, attack the now-weary victor, and they'd get the spoils. The same balance works the other way if the Arabs started the war. They'd be defeated by a fresh army, and a client state set up in northern arabia.
Whereas taking the Arabian Peninsula... that could work, but again - you'd probably have a northeast/southwest split, with client states in the desert, and a return to a two-power system where the wars could return.
I can see wars to install friendly dynasties perhaps, but otherwise, no - I don't see 1984 eternal war between three enemies. The reasoning in the book was to justify the existence of the states, not just war for its own sake. Otherwise, no-one has any benefit, except in an 'Arabian Campaign' to return to a 2-power scenario.
However, the easiest ways for it to fall apart is one of the three realms getting an idiotically xenophobic or zealotic ruler, but : -
1) The Sassanids look ready to be replaced by a new dynasty that will need to build up its strength, and then protect against the Steppe again, they can still go east into india if the new dynasty wants a war.
2) The Romans could be more interested in conversion in the Balkans than conquests in the east that they can't secure.
2b) It now appears more advantageous for Heraclius to go west Justinian-Style if he wants a war, or north. The east and south are closed.
3) The Arabs, they have just been defeated and need to finish establishing themselves as a state and religion. They may go south and invade E.Africa, but otherwise they've just been defeated, their Holy War doesn't appear to have Allah's favour. They either need to go to war south to complete a Holy War, or find some other way to maintain their unity.
Sorry for the wall, but I figured I'd explain at length rather than just re-iterate my point.
However, I do love the idea of a Muslim East Africa
@HShafs - I didn't know that

Interesting, but if they were about when the PoD suggests Khalid and his army are destroyed, would they succeed? I'm not so sure without the early successes of Khalid. Feel free to show me wrong though.