You should get one thing straight - and this will save you a lot of headaches.
You are not driving your car from Doha to Dubai - you are importing a car from Doha to Dubai - it's quite a difference.
I did the same for my vehicle when I moved from Riyadh to Bahrain - I don't think the procedure will be that different in your case.
Firstly you have to get export plates from the motor transport Department in Doha (or whatever it's called there). You may have to pay an export tax - I was slugged for the export plates and another export duty at the border. Make sure you have all the paperwork present - if you don't they'll simply refuse to deal with you and that means a trip back to Doha.
When they let you go - you then have to face the Saudi customs - I haven't a clue what they require for the transit of export vehicles - but you can be sure they'll want proof that you're not exporting it to KSA. Also be sure to turn up at the border at the right time - they are closed on Fridays and between 3 and 6 on all other days - you'll have to wait in line with numerous truck drivers and because no one speaks English - you'll be totally in the dark as to what's required. You can be sure that if you don't have it right - it's back to the start.
On top of that I would be surprised if the Saudis rewuire a deposit (refundable of course) to ensure you don't dispose of the car in KSA.
Right - assuming you get through this hurdle - you then have to get through customs at the UAE. Once again I don't know the procedure their - but on coming to Bahrain, the car was impounded in their customs area - I had to find my way into the city - get insurance papers sorted out and then return with these before they would let me drive the car. THEN - you're given a temporary plate and asked to pay 10% of the car's value as import duty. For every day you drive the car with temporary plates you are also charged a hefty penalty fee (this ensures that you take the car to their motor officem have it inspected and get new plates (also a hefty charge) - if you get through this and still have any sanity left (AAARRRGGGHHH) you'll wonder why in bloody hell you didn't just sell the car in the first place.
Seriously though - if you are devoted to your car - I would strongly suggest you get a carrier to handle all the paperwork and hassles.

Good Luck!