I have read that Betelgeuse, known as Ardra in Hindu astrology, could go supernova. Are any of the zodiac stars or the stars of lunar mansions about to become a black hole or supernova?
I don't really know what you mean by "Zodiac stars". I suppose you mean the stars of the 12 traditional Zodiac constellations.
Alpha Scorpii, known as Antares, or Jyeshtha in Hindu astrology, is a red supergiant. It is in a similar stage of its lifecycle to Betelgeuse. It is expected to explode in a supernova in the next 10000-100000 years (the timing is not very well understood)
Other potential supernova progenitors are more distant and fainter, such as HD 168625 in Saggitarius, which is not visible with the naked eye. There is also the recurrant nova U Scorpii, a candidate for a type Ia supernova (which occurs when a white dwarf accretes a critical amount of matter) Again it is far to faint, even during an outburst, to be a naked eye star (but it would certainly be visible if it exploded in a supernova).
I assume that by "zodiac stars or the stars of lunar mansions" you're referring to visible stars that are within 8° or so of the ecliptic, since that's the ecliptic latitude range where the Sun, Moon, and planets appear.
Wikipedia has an article listing the known Milky Way supernova candidates. Some of those are currently visible stars in zodiac constellations, including Spica and Antares. Spica is not expected to go supernova for at least a million years or two. Antares could explode in the next ten thousand years or so.
There are two main causes of supernova: thermal runaway and core collapse. Thermal runaway can occur when a white dwarf (an old star that has stopped nuclear fusion) accretes a large amount of matter from a companion star, or collides with that companion.
A white dwarf star may accumulate sufficient material from a stellar companion to raise its core temperature enough to ignite carbon fusion, at which point it undergoes runaway nuclear fusion, completely disrupting it.
If we can estimate the accretion rate, or determine from their orbits how long it will take for the stars to collide with each other, we can make a rough estimate of how long it will be before the supernova occurs.
The other kind of supernova happens when a large star runs out of nuclear fuel and its core collapses.
Very massive stars can undergo core collapse when nuclear fusion becomes unable to sustain the core against its own gravity; passing this threshold is the cause of all types of supernova except type Ia. The collapse may cause violent expulsion of the outer layers of the star resulting in a supernova, or the release of gravitational potential energy may be insufficient and the star may collapse into a black hole or neutron star with little radiated energy.
Core collapse supernovae occur in stars with a mass in the range of 8 to 40 or 50 $M_\odot$ (solar masses), depending on composition.
It's much harder to estimate when a core collapse supernova will occur because we can't see the star's core. As a large star ages, it performs a series of nuclear reactions in its core. The rates of these reactions are highly dependent on temperature and pressure, and the more massive stars have higher core pressures and temperatures.
Each reaction in the series operates at a much higher temperature than the previous reaction, but it takes a long time for energy produced in the core of a star to propagate to the outer parts of the star and cause visible effects. For example, for energy produced in the solar core
the photon diffusion time scale (or "photon travel time") from the core to the outer edge of the radiative zone [is] about 170,000 years. From there they cross into the convective zone (the remaining 25% of distance from the Sun's center), where the dominant transfer process changes to convection, and the speed at which heat moves outward becomes considerably faster.
The photon diffusion time is even longer in more massive stars.
Perhaps in the future we'll be able to get more timely information about stellar core fusion processes, using neutrino telescopes, but our current neutrino detectors are far too crude for that.
For most of a star's life, it "burns" hydrogen into helium. The later fusion reactions operate on shorter and shorter time scales. For example,
a star of 25 solar masses burns hydrogen in the core for $10^7$ years, helium for $10^6$ years and carbon for only $10^3$ years. [...] the process will use up most of the carbon in the core in only 600 years. The duration of this process varies significantly depending on the mass of the star.
The following stages are even faster: neon burning and oxygen burning in a 25 $M_\odot$ star last for only a few years at most, and the final set of reactions, silicon burning, can only occur for a few days before the core collapses.
So if we knew that a star was doing carbon fusion in its core we'd be able to make a good estimate of when it's likely to go supernova. But the heat of carbon fusion simply doesn't have enough time to reach the surface of the star before the supernova happens.