Can the Falcon Heavy rocket handle sending payloads past Jupiter like the Atlas V?
SpaceX published numbers on their website near the bottom of the page.
I snapped an image to show here, since their formatting is prettier than I can do in Markdown.
You can see that it can do pretty much any of the missions.
Now you could probably do a better job with a third stage/kick stage, because while the second stage Merlin-1D Vac has a lot of thrust, its Isp is not the worlds greatest for an upper stage. (It is actually kind of sucky).
Well, it looks like it can get 141,000 lbs into low earth orbit. That is quite a bit of mass. With the right second and third stages you should be able to get enough delta-v for deep space. It isn't demonstrated capability yet, but I'd be thunderstruck if there wasn't a road map for a variety of upper stages for various payloads and targets.
ETA: Centaur is about 50,000 lbs, so it could loft one easily. There would be the usual integration issues, but raw weight isn't an issue.