At least for FEWER than 4 dimensions on the large scale, that is 1 dimension of time, and 2 dimensions of space, it can be shown from general relativity that in such 'universes' the force of gravity does not exist in a vacuum. It cannot extend outside a body and cause other bodies to feel its effects. I do not know what effects this has on the existence of stars and planets, but it is certain that such objects would look very different than any similar objects we could imagine in this universe. Without gravity, there is no known way of forming large stable bodies such as planets. Even the electrostatic force would behave very differently, and probably would not allow stable atoms to exist either.
As for a universe with more than 4 dimensions, ie 1 dimension of time and more than 3 dimensions of space, gravitational forces may exist in such universes, but so far as I understand it, the law that the force follows will not be an inverse-square law. This means that, from simple celestial mechanics, there would be no stable planetary orbits that could survive for millions of years around even a SINGLE star. Orbits would not be closed ellipses, but would be more complex and the temperature variations on the planetary surfaces would probably be far too extreme for organic processes.
The universe can, apparently, have more than 4 dimensions if it has the good sense to keep these other dimensions smaller than atomic-scale. Several modern theories that hope to unify gravity with the other forces seem to require that on the quantum scale, spacetime have a dimensionality of 10, but alternatives do exist that say that even at the quantum scale we still have only 4 physical dimensions.