Y dwarf stars have sub-zero temperatures, so I became curious how a star like these dies out and if is could go supernova.
Stars of spectral class Y are dim brown dwarfs, with temperatures of about 300-400 K (For comparison, Earth's average temperature is around 288 K, give or take, the surface of the Sun is 5,800 K, and the core of the Sun is 15 million K). Their temperature range means that they don't fuse hydrogen in the same way that "true" stars do. They may fuse deuterium ("heavy hydrogen"); Y dwarfs may not fuse lithium, as other brown dwarfs do, because they are relatively low-mass.
The fact that Y dwarfs (and brown dwarfs, as a whole) don't undergo normal stellar nuclear fusion means that they'll never undergo core-collapse supernovae, which typically happen when an extremely massive star gets to the point of iron fusion and then can no longer generate energy from fusion. Brown dwarfs work completely different; this is not at all a possibility.
Instead, they will slowly radiate away their heat until they become extremely dim, via the Kelvin-Helmholtz mechanism. For more information, see the basic example evolutionary tracks on this page; if you're really feeling ambitious, you can try going through these notes to get a better feel for brown dwarf evolution.