What happens to a white dwarf star if it has mass higher than the Chandrashekhar limit? What will happen if a white dwarf star has mass higher than the Chandrasekhar limit, i.e. 1.4 times the mass of the Sun?
 A: Let's assume you just add mass to a cold, completely degenerate white dwarf.
Now one possibility is that the added matter undergoes fusion reactions on the surface, resulting in mass loss and a Nova outburst. But let's ignore this and assume the added matter "sticks".
If we consider an ideal electron-degenerate carbon white dwarf, then its " Chandrasekhar limit" is either governed by a GR instability at finite density, or by the initiation of electron-capture reactions onto carbon nuclei. The two have very similar density thresholds $(\sim 4\times 10^{13}$ kg/m$^3$), and this sets the real Chandrasekhar limit to be about $1.39 M_{\odot}$.
If the WD mass exceeds this, then either it will collapse and initiate electron capture, or electron-capture will initiate a collapse, since electrons are being removed from the gas and it is the pressure of those electrons that prop the star up.
The final outcome will probably be a type Ia supernova, where the density and temperature in the collapsing WD become high enough to ignite carbon fusion reactions. Because the material is degenerate and degeneracy pressure does not depend on temperature, the reactions can "runaway". The amount of energy released when fusing the carbon exceeds the gravitational binding energy of the white dwarf and the whole thing explodes leaving no remnant (a back-of-the-envelope calculation is here).
An alternative idea is that electron capture proceeds so rapidly that there is insufficient time for enough energy to be liberated by carbon fusion to blow the star part (e.g. Metzger et al. 2009). The outcome then will most likely be an "underpowered" supernova that leaves behind a neutron star remnant. This is known as "accretion-induced collapse". This scenario is more likely for a white dwarfs that exceed the mass limit by merger with another white dwarf or by accretion onto a white dwarf with a greater oxygen content, because oxygen is harder to ignite than carbon (e.g. Tauris et al. 2013).
A: According to Wikipedia
The Chandrasekhar limit  is the maximum mass of a stable white dwarf star. The limit was first published by Wilhelm Anderson and E. C. Stoner, and was named after Subrahmanyan Chandrasekhar, the Indian-American astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 19.
White dwarfs with masses greater than the limit undergo further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole.
Also read this nice lecture.
A: Above the Chandrashekhar limit you would obtain a neutron star: https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit
Also see: https://en.wikipedia.org/wiki/Chandrasekhar_limit
