This page by physicist John Baez explains what will happen in the long term to bodies that aren't massive enough to collapse into black holes, like rogue planets and white dwarfs, assuming they don't cross paths with preexisting black holes and get absorbed. Short answer: they'll evaporate, for reasons unrelated to Hawking radiation. It's apparently just a thermodynamic matter, presumably due to the internal thermal energy of the body periodically causing particles on the surface to randomly get enough kinetic energy to achieve escape velocity and escape the body (the wiki article here mentions this is known as 'Jeans escape'). Here's the full discussion:
Okay, so now we have a bunch of isolated black dwarfs, neutron stars, and black holes together with atoms and molecules of gas, dust particles, and of course planets and other crud, all very close to absolute zero.
As the universe expands these things eventually spread out to the point where each one is completely alone in the vastness of space.
So what happens next?
Well, everybody loves to talk about how all matter eventually turns to iron thanks to quantum tunnelling, since iron is the nucleus with the least binding energy, but unlike the processes I've described so far, this one actually takes quite a while. About $10^{1500}$ years, to be precise. (Well, not too precise!) So it's quite likely that proton decay or something else will happen long before this gets a chance to occur.
For example, everything except the black holes will have a tendency to "sublimate" or "ionize", gradually losing atoms or even electrons and protons, despite the low temperature. Just to be specific, let's consider the ionization of hydrogen gas — although the argument is much more general. If you take a box of hydrogen and keep making the box bigger while keeping its temperature fixed, it will eventually ionize. This happens no matter how low the temperature is, as long as it's not exactly absolute zero — which is forbidden by the 3rd law of thermodynamics, anyway.
This may seem odd, but the reason is simple: in thermal equilibrium any sort of stuff minimizes its free energy, E - TS: the energy minus the temperature times the entropy. This means there is a competition between wanting to minimize its energy and wanting to maximize its entropy. Maximizing entropy becomes more important at higher temperatures; minimizing energy becomes more important at lower temperatures — but both effects matter as long as the temperature isn't zero or infinite.
[I'll interrupt this explanation to note that any completely isolated system just maximizes its entropy in the long term, this isn't true for a system that's in contact with some surrounding system. Suppose your system is connected to a much bigger collection of surroundings (like being immersed in a fluid or even a sea of cosmic background radiation), and the system can trade energy in the form of heat with the surroundings (which won't appreciably change the temperature of the surroundings given the assumption the surroundings are much larger than the system, the surroundings being what's known as a thermal reservoir), but they can't trade other quantities like volume. Then the statement that the total entropy of system + surroundings must be maximized is equivalent to the statement that the system alone must minimize a quantity called its "Helmholtz free energy", which is what Baez is talking about in that last paragraph--see this answer or this page. And incidentally, if they can trade both energy and volume, maximizing the total entropy of system + surroundings is equivalent to saying the system on its own must minimize a slightly different quantity called its "Gibbs free energy" (which is equal to Helmholtz free energy plus pressure times change in volume), see "Entropy and Gibbs free energy" here.]
Think about what this means for our box of hydrogen. On the one hand, ionized hydrogen has more energy than hydrogen atoms or molecules. This makes hydrogen want to stick together in atoms and molecules, especially at low temperatures. But on the other hand, ionized hydrogen has more entropy, since the electrons and protons are more free to roam. And this entropy difference gets bigger and bigger as we make the box bigger. So no matter how low the temperature is, as long as it's above zero, the hydrogen will eventually ionize as we keep expanding the box.
(In fact, this is related to the "boiling off" process that I mentioned already: we can use thermodynamics to see that the stars will boil off the galaxies as they approach thermal equilibrium, as long as the density of galaxies is low enough.)
However, there's a complication: in the expanding universe, the temperature is not constant — it decreases!
So the question is, which effect wins as the universe expands: the decreasing density (which makes matter want to ionize) or the decreasing temperature (which makes it want to stick together)?
In the short run this is a fairly complicated question, but in the long run, things may simplify: if the universe is expanding exponentially thanks to a nonzero cosmological constant, the density of matter obviously goes to zero. But the temperature does not go to zero. It approaches a particular nonzero value! So all forms of matter made from protons, neutrons and electrons will eventually ionize!
Why does the temperature approach a particular nonzero value, and what is this value? Well, in a universe whose expansion keeps accelerating, each pair of freely falling observers will eventually no longer be able to see each other, because they get redshifted out of sight. This effect is very much like the horizon of a black hole - it's called a "cosmological horizon". And, like the horizon of a black hole, a cosmological horizon emits thermal radiation at a specific temperature. This radiation is called Hawking radiation. Its temperature depends on the value of the cosmological constant. If we make a rough guess at the cosmological constant, the temperature we get is about $10^{-30}$ Kelvin.
This is very cold, but given a low enough density of matter, this temperature is enough to eventually ionize all forms of matter made of protons, neutrons and electrons! Even something big like a neutron star should slowly, slowly dissipate. (The crust of a neutron star is not made of neutronium: it's mainly made of iron.)
