It could go either way.
On the one hand, there’s nothing in canon to suggest that the Time Turners could go forwards in time. They’re only explicitly used to travel backwards in time (in PoA, methinks), and they also reversed the age of the Death Eaters in the Dept. of Mysteries. It also causes hideous complications, and is probably much riskier, and creates paradoxes. Bit of a reverse Grandfather paradox.
Consider: you travel forward in time, learn that you die in some preventable way. Back in the present, you make sure that you don’t die. But now you were never able to see yourself die, so how did you know to prevent your death? It gets complicated.
Jumping out-of-universe, I also think it would be difficult to do within the books. It would obviously tell us what was going to happen, but unless it was done in an extremely clever way, it would probably tell us too much information that we might not want to know. (Spoilers!) It could certainly make a nice read, but I don’t think there’s much JKR could accomplish by doing it.
Perhaps you could go forward in time, to predict Voldemort’s moves? But then did nobody use it the first time he was in power? (Suppose it wasn’t invented/developed then.) Go backwards to save the Potters, the Longbottoms, Dumbledore and Sirius? It’s an incredibly powerful weapon: if only one side had it, the war would be over in seconds. You also open up complex moral dilemmas. For example, we travel forwards and learn that a hitherto unknown Death Eater commits a vicious murder. Can we punish him in the present for the crime we know he’ll commit, if the punishment means he won’t ever commit said crime? And other such complications.
So there’s no explicit reference to Time Turners having forward-going abilities in canon, and even if there were, JKR probably wouldn’t have used it.
I think it should be possible. We know from OotP that the Ministry were doing research with the Time Turners, and probably into time travel itself. It's plausible that they tried to make the Time Turner travel forwards, as well as backwards. Both as an experiment in intellectual curiosity, and to ensure that they could do it if somebody else like Voldemort managed to (Cold War-esque precaution). (You could conversely argue that they’d have shied away from it, to avoid it being stolen. Let nobody have the power to travel forwards.)
As a slight digression into real-world physics, we know that wizards and witches have ultra-fast travel. In the real world, there's a phenomenon called time dilation. In a nutshell, if you're travelling really fast, you move through time ever so slightly faster than everybody around you.
The classic example is the twin paradox. Suppose Fred hops on his broomstick and travels around the universe for what he thinks is forty years at near the speed of light. When he returns, he’ll be forty years older than when he begun. But George will have aged by eighty years, because time moved about twice as fast for him as it did for George.
So if wizards can travel quickly (apparition, portkeys or the Floo network), it’s conceivable that they could have near light speed travel (even superluminal), and graft that effect onto Time Turners (or use it some other way). That allows them to travel forwards, but then they have no way of travelling back. Hmm…
Zapping the Time Turner with a powerful reversing spell might do the trick. Other ideas are scattered across the web; a search for time travel in Harry Potter will turn up plenty of stuff.
In summary, the canon doesn’t give us an answer either way, but it’s possible to make a case either way. Personally, I think a Time Turner could be used to travel forwards (albeit at great difficulty), but there’s no definitive yes/no answer.