Gulf states would switch to a financial economy as quick as possible, states such as the UAE and Qatar would succeed but states such as KSA and Bahrain would fail. In the immediate short term nothing would happen but within a year there would be widescale protests through out KSA as public sector employs realise they have no equity what so ever.
Outside of the gulf, North African countries would undergo massive social upheaval as the ruling elite and powerclass fled and left behind those who are unforunate enough to not be able to escape. The Levant would be sheltered primarily due to their lack of reliance on petrochemical dollars though Syria would degenerate.
Iran would diversify and overcome the situation. They've been under embargo one way or another for nearly 30 years so they have experience in maintaing the necessary cash flow to pay their citizens and avoid major economic collapse.
In the medium term you would see the complete collapse within the gulf of the ruling class. The royal families would flee and take as much money as they could while their underlings would be handed the reigns. They would fail and social anarchy would take over before the common people reverted back to Islamic governance. For a generation at least you would see fundamentalist Islamic theocracies pop up as people vied for power but no real governing institution would rule.
Now that we're looking so far into the crystal ball things could go either way but there is a possibility that the Arabs would return to a Kilafa style of governance which would actually reunite the Arab World, see human rights (at least an Islamic interpretation of it) flourish, education take off, economic and social reform at a level never thought possible etc etc.
But really, at the end of the day shutting the oil off will probably only have minor consquences. The Middle East has spent trillions of dollars diversifying to protect themselves from the inevitable reduction in demand for oil.