Just (Nov - Dec - Jan) spent 10 weeks in Iran with a diesel car. No problem entering the country; I've heard in the past some charges for diesel vehicels, though these may be for people without carnets or with transit visas. Be prepeared for anything at the border, they asked me for US$100 insurance, but I made a big fuss and waved my arms around and I left without shedding a dime. They may give you some transit document stating your route and tell you that you cannot stray from it. Mine gave me 7 days.... but I 'lost' it. Whenever the police stopped me (and they frequently did), my passport, and sometimes my carnet were always enough to get me free. You should technically get transit plates, but nobody said anything to me at the border, and the police always gave up haranguing me after a few minutes of me shouting TOURIST at them.
If you see you are being followed by a car, it's probably (overtly obvious) undercover police, but they're far more interested in your safety than busting you for being a spy etc.
Diesel can be found at big service stations, and is stupidly cheap (16.5 tomans = US $ 0.015), but there are not many service stations, so fill up every time you see one. Bandar Abbas has only ONE station, down near Shah Rajahee docks.
As you head east towards Pakistan, you may be asked for some permit to buy diesel. I just gave the guy my passport number, and he let me take 165 litres before I entered Pakistan, but maybe he was just nice (almost all Iranians are).
Always check your change at the pumps, try to fill up with a round amount. And take some gloves, because diesel pumps are invariable FILTHY, leaking old things. They don't stop when the tank is full, so stand to one side and wait for a fountain!
The government is planning to ration diesel (a crazy move given the amount of lorry traffic), and it could happen at any time. They gave about 5 hours notice before announcing petrol rationing, though they had already issued the cards. I don't believe any such cards exist for diesel as yet.
One final thing, the traffic in Iran is the worst I have ever seen, worse than India, Pakistan or Lebanon. The small cities are the worst, with mindless idiots pulling out from all sides without looking. Be very careful. If you don't buy insurance at the border, get it at a major city unless you like to gamble. Remember that blood money for killing a person is the rule in Iran, and it's 40,000,000 tomans at the moment, about $30,000. This isn't India.
Take care and enjoy Iran, it's 100% amazing.