Due to the huge number of stars in the universe, will there ever be a time that the night sky is filled up completely with stars such that the night sky is as bright as it is in the daytime?
The fact that the night sky is not currently filled up with stars is due to the finite speed of light, so that the light of very distant stars hasn't reached Earth yet. However, the universe is expanding, and that expansion is accelerating. This expansion is not bound by the cosmic speed limit of the speed of light and can, in fact, expand faster than light. Thus as the universe continues to accelerate its expansion, at some point the expansion of the regions of space occupied by very distant galaxies will, relative to us, be faster than light and thus the light from those galaxies will never reach us.
Your question is very much along the lines of Olbers' Paradox, and I encourage you to research that further (and specifically the paradox's resolutions) if you want to know more.
I think the best answer as far as we know, is dissipation of light energy. The electromagnetic phenomena known as light, is a wave of alternating electric and magnetic fields. It is a 3 dimensional wave, so if you have a point source the wave travels outwards like ever increasing spheres. The surface of the spherical wave gets larger and larger. The light loses its intensity over distance, eventually becoming weak enough it is no longer detectable. Imagine how intense the sun appears from mercury compared to Saturn or Pluto.