Question

How many observers would it take to cancel out the number of photons produced in a medium sized star like our Sun, thus making it dark?

(I want to consume more photons than the star is generating)

Answer 1

It is certainly true that observation of a photon destroys it. However the energy of the photon has to go somewhere, and it usually goes into heating up the observer.

Suppose you surrounded the Sun with a complete sphere of observers, then very quickly the shell of observers would heat up to the same temperature of the Sun and would start emitting photons itself. Then result would be that distant observers would still see the same net radiation coming from the Sun.

In a sense this already happens. The energy the Sun produces is generated by a relatively small part of the Sun near its centre. The rest of the Sun absorbs photons heading outwards then emits photons in turn. It takes energy emitted from the core at least 100,000 years to escape from the surface. I say "at least" because estimates of the energy travel time vary widely. Anyhow, if you count the outer layers of the Sun as "observers" then the process you describe already happens.