You are a little confused in your stellar evolution model. After the ignition of hydrogen fusion in the core of a star, it will next progress to helium fusion, then to carbon/oxygen fusion via the triple-alpha process (I've skipped a lot of steps and details there, if you want the details you can look at either Hansen & Kawaler's Stellar Interiors text or Dina Prialnik's Introduction to Stellar Structure text). What happens next is mass-dependent (using $M_\odot\simeq2\cdot10^{33}$ g and the mass of the star as $M_\star$):
- $M_\star\gtrsim 8M_\odot$
- able to continue fusion in the core
- will later blow up in core-collapse supernova events, producing either a neutron star or a black hole (mass-dependent) after forming iron in the core
- $M_\star\in(\sim0.5,\,\sim8)M_\odot$
- unable to continue fusion in the core due to insufficient temperatures
- will proceed into the planetary nebula phase (which has nothing to do with forming planets, but it's discoverer, William Herschel, thought that it was a planetary system forming)
- these stars form the white dwarfs that the Chandrasekhar limit applies to
- $M_\star\lesssim0.5M_\odot$
- unable to produce helium in the core (insufficient temperatures)
- expected to continue burning hydrogen for $t_{burn}>t_{age\,of\,universe}$
Thus, not every star produces iron in the core; this only applies to stars with mass $\gtrsim8M_\odot$.
The Chandrasekhar limit arises from comparing the gravitational forces to an $n=3$ polytrope (see this nice tool from Dr Bradley Meyer at Clemson University on polytropes)--polytropes basically mean $P=k\rho^{\gamma}$ where $P$ is the pressure, $k$ some constant, $\rho$ the mass density and $\gamma$ the adiabatic index.
That is, in order to find the limit, you need to use the hydrostatic pressure,
$$
4\pi r^3P=\frac32\frac{GM^2}{r}\tag{1}
$$
and insert the pressure of the polytrope of index $n=3$ (requires numerically solving the Lane-Emden equation) and then solving (1) for the mass, $M$. If you've done it correctly, you'll find $M_{ch}=1.44M_\odot$.
We think that most neutron stars are produced in the cores of massive stars and result from the collapse of a core that is already at a mass of $\sim 1.1-1.2 M_{\odot}$ and so as a result there is a minimum observed mass for neutron stars of about $1.2M_{\odot}$ (see for example Ozel et al. 2012). Update - the smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015).
The same paper also shows that there appears to be a gap between the maximum masses of neutron stars and the minimum mass of black holes.
You are correct that current thinking is that the lower limit on observed neutron star and black hole masses is as a result of the formation process rather than any physical limit (e.g. Belczynski et al. 2012 [thanks Kyle]).
Theoretically a stable neutron star could exist with a much lower mass, if one could work out a way of forming it (perhaps in a close binary neutron star where one component loses mass to the other prior to a merger?). If one just assumes that you could somehow evolve material at a gradually increasing density in some quasi-static way so that it reaches a nuclear statistical equilibrium at each point, then one can use the equation of state of such material to find the range of densities where $\partial M/\partial \rho$ is positive. This is a necessary (though not entirely sufficient) condition for stability and would be complicated by rotation, so let's ignore that.
The zero-temperature "Harrison-Wheeler" equation of state (ideal electron/neutron degeneracy pressure, plus nuclear statistical equilibrium) gives a minimum stable mass of 0.19$M_{\odot}$, a minimum central density of $2.5\times10^{16}$ kg/m$^3$ and a radius of 250 km. (Colpi et al. 1993). However, the same paper shows that this is dependent on the details of the adopted equation of state. The Baym-Pethick-Sutherland EOS gives them a minimum mass of 0.09$M_{\odot}$ and central density of $1.5\times10^{17}$ kg/m$^3$. Both of these calculations ignore General Relativity.
More modern calculations (incorporating GR, e.g.
Bordbar & Hayti 2006) get a minimum mass of 0.1$M_{\odot}$ and claim this is insensitive to the particular EOS. This is supported by Potekhin et al. (2013), who find $0.087 < M_{\rm min}/M_{\odot} < 0.093$ for EOSs with a range of "hardness". On the other hand Belvedere et al. (2014) find $M_{\rm min}=0.18M_{\odot}$ with an even harder EOS.
A paper by Burgio & Schulze (2010) shows that the corresponding minimum mass for hot material with trapped neutrinos in the centre of a supernova is more like 1$M_{\odot}$. So this is the key point - although low mass neutron stars could exist, it is impossible to produce them in the cores of supernovae.
Edit: I thought I'd add a brief qualitative reason why lower mass neutron stars can't exist. The root cause is that for a star supported by a polytropic equation of state $P \propto \rho^{\alpha}$, it is well known that the binding energy is only negative, $\partial M/\partial \rho>0$ and the star stable, if $\alpha>4/3$. This is modified a bit for GR - very roughly $\alpha > 4/3 + 2.25GM/Rc^2$. At densities of $\sim 10^{17}$ kg/m$^3$ the star can be supported by non-relativistic neutron degeneracy pressure with $\alpha \sim 5/3$. Lower mass neutron stars will have larger radii ($R \propto M^{-1/3}$), but if densities drop too low, then it is energetically favorable for protons and neutrons to combine into neutron-rich nuclei; removing free neutrons, reducing $\alpha$ and producing relativistic free electrons through beta-decay. Eventually the equation of state becomes dominated by the free electrons with $\alpha=4/3$, further softened by inverse beta-decay, and stability becomes impossible.
Best Answer
First, I want to make clear precisely what the Chandrasekhar mass is: the maximum mass of a white dwarf supported purely by electron degeneracy. It depends on a few things (notably, the mean molecular weight of the white dwarf) and neglects other sources of or deviations to pressure, but its canonical value of 1.44 $M_\odot$ is a pretty accurate estimate at the mass above which a white dwarf or degenerate stellar core collapses.
The next possible source of support against gravity is neutron degeneracy. i.e. support from the fact that you can't squeeze neutrons into the same quantum state. The Tolman–Oppenheimer–Volkoff limit is the corresponding mass limit for the maximum mass of an object supported by neutron degeneracy. It is much more difficult to calculate this maximum mass because it requires precise knowledge of the equation of state and, currently, we just don't know exactly how matter behaves under those conditions. Even so, the limit is probably more than 2 $M_\odot$ because such a neutron star has been observed and broadly thought to be less than about 3 $M_\odot$. The upper end of the range is very rough, though.
Black holes, however, have no such mass limit because there is no pressure support. We are quite confident that there are black holes at the centres of distant galaxies with masses that exceed 10$^{10}$ $M_\odot$, and nearby M87 hosts a black hole of a few 10$^9$ $M_\odot$. Any apparent limit on the mass of a black hole is just because it hasn't been fed enough.