Today is the scheduled launch date for the Falcon Heavy demonstration. As of this writing, the launch is on schedule for an 11:30 MST launch.
There is very little official guidance from SpaceX as to what to expect. Elon Musk has stated that minimum mission success is clearing pad 39A far enough such that any further failure doesn't destroy that pad. There has never been a catastrophic failure at pad 39A, and Musk would like to keep it that way.
However, the plan is to do a boost, then three burns of the upper stage. The first finishes launch to LEO. The second is about 30 seconds, and seems to put the booster into a GTO-like orbit. Lifting a heavier spacecraft into full GTO takes about a minute, so there is some hint that this will go into an elliptical orbit that is short of GTO. Part of the demonstration is a 6-hour coast. They are doing it on this flight because the upper stage is very similar to any normal Falcon 9 upper stage, and any demonstration on this stage would apply there. This coast is what would be needed for a 3-burn GSO direct insertion, that apparently is very interesting to the military. For one thing, it would demonstrate that the upper stage could put a GPS satellite directly into its target orbit, like the much more expensive Delta IV medium. A bit more oomph and a similar endurance would put a spacecraft directly into GSO.
In any case, the consensus on NasaSpaceflight is that the target high orbit is one with a period of around 6 hours. After this coast, the second stage would be back at perigee, ready to take maximum advantage of the Oberth effect.
SpaceX has claimed that they will put the payload (A cherry-red Tesla Roadster) into an "earth-mars heliocentric orbit". The launch window for Mars is in May, so they will be launching 3 months out of the window, but since this is such a light payload, they should have plenty of C3 and probably could target Mars if they wanted. However, I think that they will instead target an orbit with periapse at Earth and a C3 typical of launching to Mars. The payload will reach the vicinity of Mars orbit, but Mars will be far far away by then. In fact, to be responsible about Planetary Protection, they should launch into an orbit which will not actually intersect Mars orbit at all, so that there is never any possibility of the car impacting Mars.
Running the numbers based on the Trajectory Planner 1.1.1 from Orbit Hangar, I get a departure C3 of 23.9 km^2/s^2, with a departure today and an arrival on October 17, 2018. The flight time is 252 days. This C3 is high for a Mars launch, but should be doable with such a light payload.
If they are targeting Mars, then the launch vehicle must be able to adjust azimuth in order to target Mars at any time during the window. If they are just going for a given C3, they can use the same azimuth whenever they go. Since ASDS is parked somewhere definite to catch the core stage, I estimate that they are targeting a fixed azimuth independent of launch time.
There are no signs of high-gain antennas or solar panels on the payload, so it is almost certain that once the battery runs down, the payload will become inert. However, the payload is an electric car, with many many amp-hours of battery life. The car radio might run for hours or days.
