Okay, let's say that every single other star in the universe vanishes suddenly.
People on earth would not notice for years. Light would still be coming from those stars, because it was currently in transit when the star vanished. We could not even tell for years, because most scientists believe gravity also travels at the speed of light (at least according to The Elegant Universe. But because these stars are so far away, the difference in gravity (when we do notice it) would be minimal. Their gravitational fields at a distance of so many million miles would be infinitesimal.
What other influences can they have? Well, light. You'll have to remember that these stars are far enough away that the light from them must have a tiny effect on Earth. They look dim enough in the first place; it would be like trying to measure the effect of a flashlight shined from the space shuttle. There simply would be no difference.
Animals really don't care about the stars. The only difference would be those that use stars to navigate.