When did the concept of "celestial sphere" fall? I'm just done reading The Great Ptolemaic Smackdown, and it provided a good timeline of establishing heliocentrism, broad acceptation of Kepler's system, and the initial (failed) Galileo's experiment attempting measurement of parallax of distant stars (which was based on assumption of two near stars of different brightness being a long distance apart; Galileo unfortunately picked a binary star) - all the models described within assume a fixed, mostly immutable "celestial sphere" of distant stars.

When, and how was this concept disproven?
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.