If a star were to suddenly lose nearly all of its stored heat, would it be able to return to its normal state? If not, would it be possible to approximate the maximum heat energy a star could lose before the change became irreversible?
If you suddenly sucked all the heat out of a normal star like the Sun, nuclear reactions would cease, the gas pressure would fall to zero. The star would collapse, initially on a freefall timescale (about 1 hour for the Sun).

The released gravitational potential energy would half be radiated away and half would go into re-heating the interior. The temperature would rapidly rise, the collapse would be halted, nuclear reactions would restart, the star would slowly expand again and a thermal and hydrostatic equilibrium should be restored on a Kelvin-Helmholtz timescale (about 10 million years for the Sun).

One way to see this, is to compare the current thermal content of a star with its capacity for generating new heat by nuclear reactions. e.g. approximate the Sun as a uniform ball of ideal gas at an average temperature of a few million degrees. The thermal content is roughly $10^{41}$ J. The Sun generates this much energy in nuclear reactions in only 8 million years - coinciding with the timescale to reestablish an equilibrium. Or you could compare it with the gravitational potential energy of around $2\times 10^{41}$ J. i.e. There is enough gravitational potential energy to reheat the Sun.

However, not everything would be a continuation. The collapse and reheating would be similar to the star repeating its pre main sequence phase. During this phase the star becomes fully convective and so the core material would get completely mixed with the rest of the star again. This would have the effect of rejuvenating the star as a new zero age main sequence star, but with a slightly higher helium abundance.