Shift in demand or supply? It depends on who pays for the petrol: Khaleej Times
Economic principles

The price of petrol is increased. Petrol and cars are complements. The demand for rental cars declines. The price of rental cars declines:

Describing the renting rates of small cars, Hani said, “At present, I charge Dh70 for a Toyota Corolla 1993 model, but I am going to reduce it to Dh65. Because the petrol is the responsibility of the client, so I will reduce the car rent so as to make it affordable,” he said.

On the other hand, the story is different in the case where the company includes petrol with the rental. Petrol is an input to production. An increase in the price of petrol increases the rental company costs and reduces supply. The result is an increase in the rental price the consumer pays:

But for customers of companies leasing out bigger cars including limousines and also offering chauffeur services, prices will see an upward trend. “At present, we’re not going to increase the rents because the cars are pre-booked,” said Mahmoud Akram of Auto Plus. “But from next month, the hiring charges for chauffeur driven cars — that are rented with fuel charges included – will definitely go up Dh100-200,” he said. “Because the petrol will now cost us an extra Dh70 for bigger cars, we will charge it to the customer and let them know about it,” he added.

Notice that small cars rent by the day, week and month, whereas the larger cars rent by the hour. This may explain the difference in business practices regarding whether the customer or the company pays for the petrol. You can only go so far in an hour. But in a day your driving behavior will differ depending on whether you are paying for the petrol or not. You and the company are better off if you internalize the cost of going an extra kilometer.

UPDATE. See also this 18Sep Khaleej Times article about the UAE rental car market. Quoting:

"I am almost out of business. I have put some of the cars for sale and have dismissed several employees. Some companies, that were tremendously affected, have liquidated their businesses to avoid further losses. I have heard rumours of further petrol hikes. If this is true, it will definitely be a fatal blow to our business,'' Abdullah Rashid, a depressed owner of a car renting company declared.

"The rise in petrol prices have put us in hot water. During the past years, our business was flourishing, but now we are cornered and there is no way out. I had bought the cars available in my company by a bank loan, which I am currently unable to pay. This resolution should not have been applied suddenly and on the spot by the government. We should have been notified a few months earlier, in order to face the unpleasant dilemma and work out a solution to this catastrophe,'' another flummoxed owner of rent-a-car asserted.