It is said that neutrinos can travel for billons of years at almost the speed of light without interacting with anything. That being the case,what do they do - is it possible they could orbit the universe indefinitely? They certainly can't leave it.
It depends on the ultimate fate of universe, as with everything else in the universe. Fundamentally, neutrinos are not much different from “ordinary matter”. Yes, they do interact only via weak force and gravity, but this only quantitatively changes the frequency of interaction, but does not change the game qualitatively. As long as there’s non-zero chance for them to interact, and as long as there is non-zero density, there is non-zero probability they will eventually interact with another particle (or black hole), and given infinite timeframe, it will happen for sure.
Accelerating expansion of universe does however complicate things a bit, but again, eventually there is bound to be some random fluctuation which will knock the neutrino off from its track.
Eventually neutrinos would meet anti-neutrinos and annihilate each other. Since the chances for this are really low, I would suspect neutrino-antineutrino collisions being the very last blink of this universe - given heat death is the ultimate fate of this existence.