Given that there are a finite number of photons that exist in the universe, and the maximum speed of a photon is governed by the speed of light, is it possible that enough people observed an object, there would not be enough photons for them all to be able to perceive the entire object and parts of it would seem to disappear?
Let say we have a candle. There are photons emitted from the candle in all directions, a finite number of photons per unit of time. If an observer is far enough away from the candle, the observer will only occasionally see (detect) a photon from the candle.
So as far as I can understand the question, this is the type of situation referred to. At a far enough distance from a source (star, candle, firefly) a photon might or might not be detected by a given observer during a given time interval. Yes, in this sense, the source could seem to disappear, because there will be intervals of time during which no photon is detected from the source.
Yes, but it happens all the time.
If other observers have absorbed all the photons which would otherwise have reached you, then they have blocked your view.