Some say we live in the golden age of the universe because there exits countless number of stars that shines in the dark universe. As the supply of gas for star formation is steadily being exhausted, it is estimated that star formation will cease in 100 trillion years.
Question: Will there be a day that the universe becomes completly dark when all the stars burn out? Does all the cooled bodies eventually collide due to gravitational forces? Will black holes eventually dominate the universe?
I've listed some references below. The most useful general review paper is Adams 1997, except that it predates the discovery of dark energy. There is also a Wikipedia article.
Will there be a day that the universe becomes completly dark when all the stars burn out?
Yes. The hydrogen fuel burned by main-sequence stars only decreases over time and is never replenished. Some hydrogen will end up permanently unavailable to star formation, e.g., in gas giants. The last stars to burn will probably be very small, dim, "frozen stars" that can exist only because of the high proportion of heavy elements (Adams, p. 8). By $10^{14}$ years from now (possibly sooner), all star formation will have ceased and all stars will have evolved into degenerate objects (Adams, p. 9).
Does all the cooled bodies eventually collide due to gravitational forces?
No. Gravity doesn't typically make things collide, it makes them orbit each other. On time scales of $10^{19}$ yr, most stars will be ejected from the galaxy (Adams, p. 12). (I found this counterintuitive due to conservation of energy, but in a gravitational interaction, there's no lower bound on the negative potential energies you can achieve, so it's different from a gas of atoms.) A minority of stars will not be ejected and may either undergo random collisions (with a time scale of $10^{22}$ yr for brown dwarfs, or much longer for degenerate stars) or gradually migrate toward the galactic core on time scales of $10^{24}$ yr due to dissipation of energy into gravitational waves (Adams, p. 13). In the end, about 1-10% of stars will end up eaten by the central black hole, while the rest escape the galaxy (Adams, p. 17).
Will black holes eventually dominate the universe?
No. As described above, most stars end up as brown dwarfs, white dwarfs, or neutron stars, which are ejected from their galaxies. A body like a brown dwarf will gradually lose its atoms to the interstellar medium. Due to some counterintuitive thermodynamics, these atoms are probably eventually spontaneously ionized (Baez 2004). Baez gives a general argument that takes into account the cosmological environment, but to get the basic idea, I like the following argument given by Peierls 1979. Take a gas of hydrogen atoms. The sum $Z=\Sigma_{n=0}^\infty e^{-\beta E_n}$ diverges, so in the limit of low concentration, where $n$ can go arbitrarily high, the probability of any discrete state is zero. Although the temperature is also going down over time, it reaches a finite limit, which is set by the Hawking radiation associated with the cosmological horizon.
This ionization turns our dead stars into a population of unbound massive particles, which adds in to the population of such particles that simply never happened to undergo gravitational collapse into a macroscopic body. (If proton decay exists, then it modifies this picture somewhat, e.g., neutron stars evolve in certain ways, but the end result should be the same.)
In addition to these particles, we have a population of black holes. On sufficiently long time-scales, these evaporate into a variety of particles, the most numerous of which are photons (but every possible type of particle is created by Hawking radiation).
So we now have a universe whose only inhabitants are various individual particles: photons plus massive particles. As the universe expands by a scale factor $a$, the mass-energy density due to photons falls off as $a^{-4}$, while the mass-energy density due to material particles goes like $a^{-3}$. The differece in exponents is because photons get cosmologically red-shifted. This will cause the photons to eventually become a negligible component in terms of their contribution to the mass-energy density.
Accelerating cosmological expansion causes the massive particles (probably mostly dark matter, neutrinos, and electrons and positrons) to end up within their own cosmological horizons, so they can no longer interact.
You can get some variation in the above story if you make unusual assumptions about the equation of state for dark energy. Baez, for example, seems to be implicitly assuming that dark energy acts like a cosmological constant, which is the most conservative interpretation right now. But, e.g., it's possible under other assumptions to have a "big rip" scenario.
There seem to be a huge number of people on the internet who believe that the universe of the far future will consist of nothing but photons, since all matter will have been recycled through black holes and Hawking radiation. The above analysis shows that this is simply untrue, but this folk belief seems to have the same kind of grip on the popular consciousness as other false factoids such as Eskimos' having $n$ words for snow or the belief that people should drink eight glasses of water a day. One reason we can be very sure that the claim about photons is not true is that Roger Penrose is a very smart guy, and he had a theory called conformal cyclic cosmology (CCC) which only seemed to be viable if he could find a way to get all matter to be recycled into photons in the distant future. This gave him the strongest possible motivation to look for mechanisms to make that happen, and after considerable (publicized) effort, he failed.
Adams and Laughlin, "A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects," 1997, http://arxiv.org/abs/astro-ph/9701131
Baez, J., 2004, "The End of the Universe.", http://math.ucr.edu/home/baez/end.html
Dyson, Time without end: Physics and biology in an open universe, Reviews of Modern Physics 51 (1979), pp. 447–460, doi:10.1103/RevModPhys.51.447.
Freese and Kinney, 2002, The ultimate fate of life in an accelerating universe, http://www.arxiv.org/abs/astro-ph/0205279
Hu, Hawking radiation from the cosmological horizon in a FRW universe, http://arxiv.org/abs/1007.4044
Krauss and Starkman, 1999, Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe, http://arxiv.org/abs/astro-ph/9902189
Peierls, Surprises in Theoretical Physics, section 3.2
It is true that all the stars will burn out, and that at some point, star formation will cease. At this point, there will still be photons travelling around in the Universe, of course, but due to cosmological expansion, these will be redshifted out in the radio and to very, very low energies, while their number density will also fall (the energy density of photons depends on the scale factor $a$ as $\propto a^{-4}$, where matter density only drops by $\propto a^{-3}$). So whether the Universe goes comepletely "dark" is a matter of definition, but it definitely goes completely dark for all practical purposes.
But the Universe expands with an accelerating expansion rate. This means that our cosmological event horizon - the limit beyond which any event happening now will never be observable, even though we might be able to still see past events from that distance - is moving closer to us in co-moving coordinates. The consequence is that all gravitationally unbound structures will eventually be isolated in each their own little patch of the Universe, beyond which they can only see history unfolding up to a certain point in the past. Depending on the rate of acceleration of cosmological expansion, this may even tear up galaxy clusters or even individual galaxies (or their burnt out chars).
Whatever the scale, we will end up with isolated islands of matter which will eventually, if massive enough, merge into black holes, possibly some supermassive ones This matter will either merge into black holes or through a long chain of decay mechanisms evaporate into freely floating electrons or positrons. But to address @udiboy 's comment above, there probably will be no new physics due to the presence of so many black holes, as their future light cones will not overlap and they cannot affect each other.
But even these black holes will also evaporate as Hawking radiation, eventually leaving a very, very dilute, gravitationally unbound soup of electrons, positrons, neutrinos and very long wavelength photons. According to this answer to a similar question, the distance between single particle will be larger than the current size of the observable Universe. The Universe will effectively be cold, dark, and almost exclusively made up of Dark Energy.
But this is just ordinary. Other contributions to the matter density are Dark Matter and neutrinos. Neutrinos make up only around $\lesssim 0.4 \%$ of the Matter-Energy density of the Universe, but while they oscillate between their flavors, they don't seem to decay into any other forms of matter, so they should still be around when all black holes have evaporated. As for Dark Matter, we don't really know how it works, so there is some uncertainty. The most commonly favored model is some kind of WIMPs (Weakly Interacting Massive Particles), which are their own antiparticles and thus can annihilate if they happen to collide, which obviously would happen very, very rarely since we don't measure any significant Dark Matter glow with our current instruments. It has been proposed (See the paper suggested by @BenCrowell above) that there could be mechanisms where White Dwarfs and Neutron Stars serve as a kind of catalyst for caption and annihilation of Dark Matter, similar to how dust grain can become catalysts for the formation of molecules in the ISM. But since the collision and hence decay rate of DM is so extremely low (if it even can self-annihilate), and given that the NS and WD catalyst would only be a passing phenomenon between everything is torn apart, I think it is safe to assume that Dark Matter will still dominate the matter density of the Universe on very long terms.
Some people (including Andrei Linde, Max Tegmark and Alan Guth) have speculated that regions in this vacuum/Dark Energy Universe, could eventually change the equations of state for Dark Energy or vacuum (if that is not the same thing) and have it decay into an inflationary field, causing new "bubble Universes" to be born in this vast, empty Universe. In fact, according to the hypothesis of eternal inflation, this is already happening all the time in an enormous, vacuum-filled Universe (or in fact a Level II Multiverse), and what we perceive as "The Universe" is only one such "inflationary bubble" out of countless, if not infinitely, many, which will eventually smooth out and become one with the vacuum it came from, and give rise to other inflationary bubbles.
