After its mission, New Horizons will be destined to travel perpetually forever. I wonder though how far it will be when the sun explodes. Will it be far enough away that when this happens, billions of years from now, it will not be destroyed? If this is the case, what will happen to New Horizons? Is it likely to literally just travel forever without ever hitting anything?
Absolutely. The death of the Sun will likely swallow an area of about Mars's radius, most likely a bit further. Thus, any spacecraft in the outer solar system or further will survive.
Also, it's likely that any spacecraft orbiting Earth will have a good chance of survival, as the Earth's orbit will be pushed back further. Mars is almost certainly going to survive somewhat.
In addition to all of this, New Horizons, and the Voyagers, etc, are all moving quite fast. Voyager 1 is moving at a rate of 3.6 au/year. The Sun won't explode for billions of years. Using 4 billion as a conservative value for that time, Voyager will be 14.4 billion au away from the sun, or basically the other side of the Milky Way (I show 227,000 light years away).
Bottom line is, the Sun's destruction will have no bearing on the lifetime of any satellite in current solar escape, and probably won't have any affect on anything in the outer solar system. Other events might have an effect, however.