Who's going to run out of money first - Iran or those two Gulf States?

You're missing the point. The core of the Gulf states' foreign policy failure is exactly that they think money is the ultimate weapon. It is not. The Gulf states have been pushing back against Iranian influence for decades now, while the latter has been under considerable financial strain from UNSC sanctions. Saudi Arabia and Qatar have had billions more than Iran at the disposal to leverage influence within the Middle East and have failed miserably.

Saudi Arabia produces no service of value. To anyone. It produces exactly one product of value, oil, and unlike, say, Russia's Gazprom, Saudi Arabia has been unable to leverage oil directly as a foreign policy weapon since 1973. For good reason; the world will not tolerate such a hamfisted threat again and, since they tried it, OPEC has been steadily undermined by internal divisions, increased oil exploration in non-OPEC countries, and most recently, a combination of renewable energy and advancement in oil exploration such as shale oil. It has reached the point where Saudi Arabia is so dependent on oil profits that any attempt to repeat the events of 1973 does catastrophic damage to the national budget. Those foreign currency reserves will not last forever, and any percentage market share, once lost, is extremely difficult to reclaim.

Qatar's main source of income, on the other hand, is natural gas. One of their main objectives has been to complete the Qatar-Turkey pipeline to establish a route to Europe. That's failed. Assad rejected it and his rule does not appear to be in danger at this point. Furthermore, the profits of Qatari gas that have gone to soft-power initiatives - al-Jazeera and the World Cup in particular - has backfired. The World Cup has not benefited Qatar's reputation; in fact it's done quite the opposite and exposed them for the corrupt, slaving autocracy that the country is. al-Jazeera has lost a crucial fight with the Egyptian military regime over their support for the Muslim Brotherhood and has been forced to back down.

My point is that Qatari and Saudi money has been misdirected from the very beginning. Both societies doesn't realise that the recipient of the money is far more important than the number on the check. Iran could spend 100 million on Hezbollah in a year and that would be more effective than Qatar dropping five times that amount to rebel groups. Money only provides the tools for an individual or group to succeed, it doesn't generate success alone. You can see this in elements of each society; Saudi Arabia's military might be equipped with the latest toys, but their officers are undisciplined idiots that can't even secure a border with Iraq. Qatar might be able to pay for skyscrapers but their society is entirely reliant on expatriate workers that possess no loyalty to (and in many cases loathe) the state.