It depends on if you want any payload on it. Using the standard Merlin 1Ds on the first stage (Merlin Vacs won't fit), and using the specifications for the Falcon 9 FT given on Spaceflight101, I compute the following delta-V figures using the rocket equation:
- 0 payload: 15645 m/s
- 10 tons payload: 11497 m/s
- 20 tons payload: 9776 m/s
(These are probably a little generous; I'm assuming all of the given propellant mass can be used, but normally engines are shut down prior to complete depletion for safety reasons.)
Situation 1: How fast could it get up to with no destination in mind and you burned all the fuel.
Adding the 7800 m/s you started with in low Earth orbit, even a 20-ton payload gets you up to 17500 m/s, comfortably higher than Earth escape velocity, comparable to New Horizons' departure speed.
Situation 2: Lets say its heading towards Mars and you wanted to leave enough fuel to get into mars orbit. Then how fast could you get there?
The upper stage of Falcon 9 has a very limited battery life, and can't keep its liquid oxygen from boiling off over that kind of journey, so it won't be able to get into Mars orbit by itself. If the payload includes a storable-fuel rocket that can do its own orbital insertion, then you can use all of the F9 to leave Earth orbit. Assuming the orbital insertion stage and payload is around 20 tons, and we're thus leaving Earth at New-Horizons-like speeds, it's about 80 days to Mars, in contrast to the usual 150 to 300 days it takes for fuel-efficient trajectories.
Note that since you’re arriving at Mars at such high speed, you’re going to need a lot of fuel to slow down into orbit, so not much of the 20 tons is going to be available for useful payload.