Probably not going anywhere near Mars
As far as I can tell, the destination is an elliptical orbit around the sun with perihelion near Earth's orbital radius and aphelion near Mars's orbital radius. As the launch is at the wrong time, when the roadster reaches aphelion Mars will be somewhere else.
The brochure for Falcon Heavy offers 16800kg to Mars. Presumably this is for an Earth/Mars transfer orbit. A 2009 Tesla Roadster is 1300kg. Even with a few hundred kg for the payload adapter a Falcon Heavy is massively over powered. A Falcon 9 can get 4000kg to MTO. There are things a Falcon heavy can and cannot do with such a light payload:
Pluto transfer orbit: The brochure offers 3500kg to Pluto.
Fast flyby of Mars. There will be an aphelion that is outside Mars orbit that puts the Roadster near Mars either on the way to aphelion or on the way back.
(Probably) cannot do orbital insertion to orbit Mars. The stage 2 engine could shut down with propellant to spare after setting up for a fast flyby of Mars. I have not seen an endurance figure for stage 2. The liquid oxygen will slowly boil away and the liquid helium will boil away more quickly. Helium is needed to pressurise the propellant to the minimum required for the pumps to operate, so the choice is to use it near Earth or lose it before you get to Mars.
SpaceX does have long endurance propulsion: Draco. Early versions of Falcon had 4 Draco thrusters on stage 2 but these have been replaced with nitrogen cold gas thrusters. A Super Draco could do something to slow down a Mars flyby, but they have 1300kg of propellant and I think we would have seen one in the pictures if they had duct-taped one onto the car.