If Heraclius moved the capital to Carthage, he'd essentially be demoting himself back to Exarch of Africa and then seceding from the Empire. If this happens at the trough of Byzantine fortunes (when Persia held the Levant, Egypt, and big chunks of Asia Minor), and especially if Heraclius were to take most of what was left of the army and the treasury with him, there's little a new Emperor in Constantinople could do to challenge him in Africa for some time.
Maybe the new Emperor in Constantinople could still scrape together enough of a force to stage a comeback against the Persians, as Heraclius did OTL, and then try to reconquer Africa. Or maybe he'd instead make an unfavorable peace with the Persians, surrendering most of the areas occupied by the Persians so he could concentrate on the Avars and hold onto what he had left.
This latter would leave Persia in a much, much stronger position against the Muslim conquests a couple decades down the road (just a few years after the OTL end of the war). I have no idea if they'd be strong enough to change the outcome of that war from OTL.