If so, what would be the direct consequences?
I see no reason to assume he had diplomatic status and I haven't heard anybody claim that he did.
Diplomatic status is usually relevant in less radical circumstances but even then there are no direct consequences. There is no overarching institutions to enforce diplomatic rules (or international law in general) and violations are simply followed by a tit-for-tat of symbolic measures from summoning an ambassador to formally breaking diplomatic relations. Obviously, the US doesn't have diplomatic relations with Iran so that's moot. The closest is for Iran to summon the Swiss diplomats in charge of the US interest section at their embassy in Teheran, which they immediately did.
But in that case, an hypothetical diplomatic status is hardly relevant. At the end of the day, Suleimany's importance to Iran was well-known to both parties. The US targeted him and targeted him because of this. They are not denying it and are threatening further action. US officials claim the attack was justified and a way to prevent further escalation and all sorts of illegitimate behavior from Iran but they are not attempting to downplay its significance. Iran is bound to treat it as a deliberate attack and to react accordingly.
Iraq is also angry for many reasons. Suleimani not having diplomatic status would not make the attack OK and, conversely, him being officially considered a diplomat would make it only marginally worse from their perspective. Ultimately, they are in a very delicate situation and will have to decide exactly how far they are willing to go in confronting one of their two powerful patrons. For now, they have summoned the US ambassador for questioning.
Third countries could in principle care a little bit about violations of diplomatic immunity (for the sake of safeguarding it in principle and protecting their own diplomats) and make some noise about it. But here again, this concern would be dwarfed by all the other interests at stake: the risk of escalation, being dragged into a war or suffering some collateral damage, security of military and diplomatic personnel in the region, seeing similar attacks on their soil, etc.