When did the concept of "celestial sphere" fall?
Friedrich Bessel's 1838 observation of stellar parallax disproved the notion that all stars were equally distant. 

However, by this time, the idea that the stars were on a sphere surrounding the solar system had already been discounted (but not disproven) by the application of the Copernican principle (The Earth does not have a privileged position) and Occam's Razor (Adopt the model that has the fewest assumptions)

Giordano Bruno is particularly noted with the idea (c. 1580) that the sun is just one star among many which are distributed throughout the universe.