ECU and Diagnostics
The 360 is controlled by an advanced Engine Control Unit and Transmission Control Unit which can log, record and can adjust engine parameters. Even with a regular OBD-II scanner, a home mechanic may not be able to diagnose fault codes as there are codes specific to Ferrari. There are some software vendors which may be able to give you some insight into your Ferrari though; owners have had success with Auto X-Ray and Auto Enginuity.
Ferrari has issued a factory recall to address problems with the variator that sits on the exhaust cam. If issues occur they can cause some major damage to the engine. A newer variator and tensior which Ferrari released have fixed this issue. Being a high performance engine, you will also need to
ensure that the oil is changed religiously according to the service schedule. Should you track your car then it wouldn't hurt to do an extra change before the maintenance interval. We've attached a quick guide below on how to do an oil change here.
Motor/Transmission Mount Wear
Due to the design of these brackets, they can crack over time in earlier models. It's wise to remove them and reinforce them with some welding. If you are not that way inclined, you can always purchase some late model 360 versions from Ferrari. Transmission mounts have a similar issue which can be fixed by upgrading to an F430 mount.
Gearbox and Clutch
The clutches in both the manual and F1 transmissions are rated to about 45,000 kilometers if treated correctly. They are definitely an expensive replacement should the car you are looking at or have is due for a clutch replacement. The gearboxes generally tend to hold up quite well, however, in a manual car it would be ideal to test changing between 3rd and 2nd to ensure that there are no cable linkage issues.