Elon Musk is quoted to have said, after the launch of Falcon Heavy, that «it can launch things direct to Pluto or beyond, no stop needed, don’t even need gravity assist or anything». Now, I am not an expert in orbital computation, but this claim seems highly dubious. New horizon took 9 years to reach Pluto. How long would it take for a spacecraft launch by Falcon Heavy without using any gravity assist?
It could have done so, but it is not the only rocket that can do so. The Atlas V that launched New Horizons to Pluto could have taken a direct path, but it would have taken 3 years longer. Falcon Heavy reportedly can lift 3.5 tonnes to Pluto directly, which is about 7 times heavier then New Horizons.
As to how quickly could Falcon Heavy have done that without the gravity assist? The exact numbers are pretty difficult to guess. SpaceX no longer publishes their performance charts like they used to. From NASA's performance estimator, I have the following charts. Note that the performance numbers are suspect, will update them when I have better.
This indicates that while a direct to Pluto mission is possible, it would not have the high energy required. Even with the expendable version, it still doesn't perform as well.
Bottom line is, the lack of a high ISP upper stage really limits the functionality of the Falcon Heavy in carrying cargo beyond Earth orbit. The mission could be done, but it would take longer then with an Atlas V 551 mission, as New Horizons used. For a fully expendable mission, the crossover point between Falcon Heavy and Atlas V 551 seems to be about $60 km^2/s^2$. That won't quite get you to Jupiter. It turns out the two are pretty equivalent in price.