How can we determine the internal structure of a star from distance? See here for the discussion leading to this question. In essence, I was wondering whether there were methods to differentiate between a star destined to become neutron star and a star destined to become a white dwarf star, whose respective masses only indicate that they could become either (if observed at some initially unknown point during their life-cycle). 
I suspect this task becomes easier in binary star systems (and potentially those with observed exoplanets), but I don't doubt that there are other, more general methods that are available to us.
 A: We don't need to "observe" a star's internal structure to know if they will end as white dwarves or neutron stars. its only a question of finding the mass of their progenitor stars.  
I think you might be confused about the Chandrasekhar limit, which only gives you the upper mass limit of the white dwarf or the lower mass limit of the neutron star. Your confusion lies in the fact that you think the 1.44 $M_{\odot}$ figure used as an upper bound for a white dwarf and as a lower bound for a neutron star is used to differentiate between progenitor stars destined to turned into either. The neutron star and the white dwarves could be understand as the "final" corpses of dead stars, not the progenitor stars you are inquiring about.
The answer to your question is that only progenitor stars more massive than  8 $M_{\odot}$-10 $M_{\odot}$ will become neutron stars. This is because only stars massive enough will have the sufficiently high temperatures and pressures in their cores to trigger Carbon and Oxygen burning inside their cores. This burning will eventually lead to a massive iron core (1.44 $M_{\odot}$ mass) that will collapse into a neutron star. 
Stars lighter than 8 $M_{\odot}$ will not burn Carbon and Oxygen and instead will shed off their hydrogen and helium layers and leave behind a cold Carbon/Oxygen star, referred as a white dwarf.  
There's also a "grey area" between  8 $M_{\odot}$ - 10$M_{\odot}$ that could either lead to a massive white dwarf or a neutron star. Such a massive white-dwarf could have burnt it's carbon layer but not have sufficient mass to burn its oxygen layer, leaving behind a massive oxygen-neon–magnesium white dwarf. 
Now, about calculating their masses. Before the stars collapse into white dwarves or neutron stars, their masses will be more or less constant once they enter the main sequence phase. So an n 8 $M_{\odot}$ star will continue to have that mass at any point of its main sequence  cycle. So the question then becomes about how to calculate their masses. There are different methods,  probably the most intuitive and simpler one is if the star is in a binary system one could use  newtonian mechanics to derive the mass. 
EDIT: Edited some stuff in lieu to Robert Jeffries comment. 
A: 
In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity. The relationship is represented by the equation:
  $$
\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^a
$$
  where L⊙ and M⊙ are the luminosity and mass of the Sun and 1 < a < 6.[1] The value a = 3.5 is commonly used for main-sequence stars.[2] This equation and the usual value of a = 3.5 only applies to main-sequence stars with masses 2M⊙ < M < 20M⊙ and does not apply to red giants or white dwarfs. As a star approaches the Eddington Luminosity then a = 1.
In summary, the relations for stars with different ranges of mass are, to a good approximation, as the following
  $$\frac{L}{L_{\odot}} \approx .23\left(\frac{M}{M_{\odot}}\right)^{2.3}   \qquad (M < .43M_{\odot})$$
  $$\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^4   \qquad\qquad      (.43M_{\odot} < M < 2M_{\odot})$$
  $$\frac{L}{L_{\odot}} \approx 1.5\left(\frac{M}{M_{\odot}}\right)^{3.5}   \qquad (2M_{\odot} < M < 20M_{\odot})$$
  $$\frac{L}{L_{\odot}} \approx 3200 \frac{M}{M_{\odot}}   \qquad (M > 20M_{\odot})
$$

This isn't me; it's Wikipedia.
