Who's saying they need to be refugees? Sure, some might be, but they could also be raiders. Again, we're talking about a massive, wealthy area in the middle of a massive power vacuum. Perfect window of opportunity for raiders to seize.
My point is that I don't think Islam necessarily determined the Arab need to push into these areas. They were very weak and juicy regardless of Islam, and it simply makes strategic and economic sense for Arab tribes to prod the Levant for weak spots.
Hejaz in my opinion has a good chance of being unified and becoming a power (though not a religious one.) Their tribes were strengthening militarily.
Yet without the universalist ideology of Islam they will ignore Najd in all likelyhood, and possibly ignore Yemen (though I see Yemen as quite likely unlike Najd, which is sort of barren.) They will be attracted to the levant and are like OTL Muslims still primed to co-opt the strength of the Ghassanid military and clans, it may even be faster than OTL as there will be no pressure on them to gradually Islamize. Likewise it will go for former lakhmid territories which would push it towards war with the Sassanids.
The Byzantines in this scenario *could* withstand the onslaught and retain Damascus, but in my opinion Persians are doomed. A less religious Arab force would make the Parthian royal houses loose interest earlier, and could be more accomodating to their local power than the Rashidun were.
That being said I don't think they will push all the way into eastern khorasan and an alexander type fragmentation is likely. The eastern successor can then focus on expanding east later, while perhaps fighting both the "western arabs" and romans for mesopotamia.
Note: I am not saying a Hejazi power is in any way inevitable. But its quite possible. It would likely be far less dynamic than the Rashidun. But its lax attitude to religion actually will have certain benefits.
Edit: Also in eastern Khorasan Sassanid remnants (whether their royal house, the parthian clans, or hepthalshahs) will have more strength and remain a force for longer, and a reconquest is not *impossible*but it would lead to something a bit different than the previous Sassanid-Parthian confederal order in Iran, even if elements of the model are retained. I do think it is either -an Arab led state subdues these eastern remnants eventually or - the remnants successfully retake Persia proper.
Mesopotamia is very wealthy real estate and Eastern/Western Arabs will fight over it probably.