A frequent source of links has sent me the coordinates of a page
This picture contains just a real-world eruption! Via IO9.
In the authors' opinion, physicists and others have a moral duty to either exclude the possibility, or look for security measures that would protect us against such a rogue regime, or prove that such a protection is impossible.
First of all, is such an explosion possible?
I don't think so. Note that the concern isn't much different from the old concern that some fathers of the H-bomb had to think about – namely whether the H-bomb would start to devour the whole atmosphere and the rest of the Earth as it realized that it's a thermonuclear fuel that may be burned.
Why is it similar?
It's similar because the present authors are afraid of the increase of the Sun's temperature from the current 9-17 million Celsius degrees in the Sun core to temperatures that are at least an order of magnitude higher and allow the reactions to be exponentially sped up. Consequently, the present temperature is negligible relatively to the desired one. In that respect, the comparison of real and "desired" temperatures is analogous to the situation on Earth where the atmosphere is also much cooler than the temperature needed to change the atmosphere to thermonuclear fuel. In both cases, the initial temperature may be neglected and approximated by zero.
Moreover, the Sun density is about 1.4 times the maximum density of water so it is surprisingly comparable to the densities encountered on our blue, not green planet, too.
The arguments showing that such a risk isn't there are somewhat subtle – the "proof" that we're safe is in no way trivial from a beginner's viewpoint. But I am confident it may be formulated. It seems to me that the "detonating Sun alarmists", much like the "detonating Earth's atmosphere alarmists", are neglecting various other quantities describing the environment that go beyond the temperature and density.
It's plausible that you may create much higher temperatures in a cubic meter of the Sun – probably only on the surface because it's implausible that any material will be able to penetrate through the solar matter whose temperature starts at 6000 Celsius degrees or so (all conventional materials melt and/or evaporate around that point). But if you create such huge temperatures in a small region, it doesn't imply that they will spread.
It seems clear to me that the rate of cooling of the "detonatingly hot" region will be too high when the size of the region grows. More importantly, you just never obtain a high density in this way. The density of the Sun is what it is, dictated by the overall solar mass and the solar volume, and an extra bomb doesn't change this counting much. Because the density of the bulk of the Sun will remain fixed, the ultimate near-equilibrium temperature of fusion is pretty much dictated by this density, and this temperature is what we observed in the Sun today.
See also my answer Why Jupiter isn't a star, whether Jupiter may be blown up and what it would mean, how to stop or blow apart a star, why asteroids around us don't have relativistic velocities, or many other questions and answers about related dramatically catastrophic cosmic (im)possibilities. ;-)
For these reasons and others, I am not really afraid of a Khamenei who would like to turn the ancient Egyptian God, the Sun, into an unhinged Allah. ;-)